Glossary

9.1 Solve Quadratic Equations Using the Square Root Property

3 min read•june 25, 2024

Quadratic equations are a key part of algebra, showing up in many real-world problems. The is a handy tool for solving these equations, especially when they're in certain forms.

This method helps us find the roots of quadratic equations, which tell us where a crosses the x-axis. Understanding this property builds a strong foundation for more complex algebraic concepts.

Solving Quadratic Equations Using the Square Root Property

Square Root Property for ax² = k

Top images from around the web for Square Root Property for ax² = k

  • Solve Quadratic Equations Using the Square Root Property · Intermediate Algebra View original

    Is this image relevant?

  • Solve Quadratic Equations Using the Square Root Property – Intermediate Algebra but cloned this ... View original

    Is this image relevant?

  • Solve Quadratic Equations Using the Square Root Property · Intermediate Algebra View original

    Is this image relevant?

1 of 3

Top images from around the web for Square Root Property for ax² = k

  • Solve Quadratic Equations Using the Square Root Property · Intermediate Algebra View original

    Is this image relevant?

  • Solve Quadratic Equations Using the Square Root Property – Intermediate Algebra but cloned this ... View original

    Is this image relevant?

  • Solve Quadratic Equations Using the Square Root Property · Intermediate Algebra View original

    Is this image relevant?

1 of 3
  • States if a2=ba^2 = b, then a=±ba = \pm \sqrt{b} (aa can be positive or negative)
  • To solve ax2=kax^2 = k:
    • Isolate x2x^2 term on one side of equation
    • Divide both sides by aa to get x2=kax^2 = \frac{k}{a}
    • Take square root of both sides: x=±kax = \pm \sqrt{\frac{k}{a}}
  • Simplify square root if possible (perfect squares, simplify fractions)
  • Solution will have two roots: one positive and one negative (±\pm symbol)
  • Examples:
    • 4x2=36→x2=9→x=±34x^2 = 36 \rightarrow x^2 = 9 \rightarrow x = \pm 3
    • 5x2=45→x2=9→x=±9=±35x^2 = 45 \rightarrow x^2 = 9 \rightarrow x = \pm \sqrt{9} = \pm 3
  • The expression under the square root sign is called the

Square Root Property with a(x - h)²

  • Equations in form a(x−h)2=ka(x - h)^2 = k can be solved using Square Root Property
  • First isolate (x−h)2(x - h)^2 term on one side of equation
  • Divide both sides by coefficient aa to get (x−h)2=ka(x - h)^2 = \frac{k}{a}
  • Take square root of both sides: x−h=±kax - h = \pm \sqrt{\frac{k}{a}}
  • Solve for xx by adding hh to both sides: x=h±kax = h \pm \sqrt{\frac{k}{a}}
  • Solution will have two roots: one where square root term is added to hh and one where it is subtracted from hh
  • Examples:
    • 3(x−2)2=75→(x−2)2=25→x−2=±5→x=2±5=73(x - 2)^2 = 75 \rightarrow (x - 2)^2 = 25 \rightarrow x - 2 = \pm 5 \rightarrow x = 2 \pm 5 = 7 or −3-3
    • 2(x+1)2=18→(x+1)2=9→x+1=±3→x=−1±3=22(x + 1)^2 = 18 \rightarrow (x + 1)^2 = 9 \rightarrow x + 1 = \pm 3 \rightarrow x = -1 \pm 3 = 2 or −4-4
  • The value of hh represents the x-coordinate of the for the parabola

Interpreting quadratic equation solutions

  • Real roots:
    • If (b2−4acb^2 - 4ac) is positive, equation has two distinct real roots
    • If discriminant is zero, equation has one real root (double root)
  • :
    • If discriminant is positive but not a perfect square, roots will be irrational (cannot be expressed as simple fraction or integer)
    • Can be left in simplest radical form or approximated as decimals (2≈1.414\sqrt{2} \approx 1.414)
  • Complex roots:
    • If discriminant is negative, equation has two complex roots
    • In form a+bia + bi, where aa is real part and bibi is imaginary part
    • ii is , defined as i2=−1i^2 = -1 (square root of -1)
  • Number of real roots provides insights into graph of quadratic function:
    • Two real roots: graph crosses x-axis at two points
    • One real root: graph touches x-axis at one point ()
    • No real roots: graph never intersects x-axis

Additional Methods and Considerations

  • The can be used to solve any in standard form (ax² + bx + c = 0)
  • When solving equations, be cautious of that may arise from algebraic manipulations
  • The graph of a quadratic equation is always a parabola, with its vertex located on the axis of symmetry

Key Terms to Review (21)

±: The symbol '±' is a mathematical symbol that represents the concept of 'plus or minus'. It is used to indicate that a value or expression can have two possible solutions, one positive and one negative. This symbol is commonly encountered in the context of solving equations and working with polynomial expressions.
Axis of Symmetry: The axis of symmetry is a line that divides a symmetric figure, such as a parabola, into two equal halves. It represents the midpoint or center of the symmetric figure, where the function changes direction from increasing to decreasing or vice versa.
Coefficient: A coefficient is a numerical factor that multiplies a variable in an algebraic expression. It represents the number of times a variable appears in a term or an equation.
Completing the Square: Completing the square is a technique used to solve quadratic equations by transforming them into a perfect square form. This method involves adding a constant to both sides of the equation to create a perfect square on one side, allowing for easier factorization and solution of the equation.
Complex Solutions: Complex solutions refer to the solutions of quadratic equations that have imaginary or non-real components. These solutions arise when the discriminant of the quadratic equation is negative, indicating that the equation has no real roots.
Constant Term: The constant term is a numerical value that does not have a variable associated with it in a polynomial expression. It is the term that remains unchanged regardless of the value assigned to the variable(s) in the expression.
Discriminant: The discriminant is a mathematical expression that determines the nature of the solutions to a quadratic equation. It plays a crucial role in understanding the behavior and characteristics of polynomial equations, quadratic equations, and their graphical representations.
Extraneous Solutions: Extraneous solutions are solutions to an equation that do not satisfy the original equation. They are not valid solutions, as they are introduced during the process of solving the equation, often through algebraic manipulations.
Factoring: Factoring is the process of breaking down a polynomial expression into a product of simpler polynomial expressions. This technique is widely used in various areas of mathematics, including solving equations, simplifying rational expressions, and working with quadratic functions.
Imaginary Unit: The imaginary unit, denoted as $i$, is a mathematical construct that represents the square root of -1. It is a fundamental concept in the complex number system, which extends the real number line to include numbers with both real and imaginary components.
Irrational Roots: Irrational roots refer to square roots, cube roots, or other roots that result in values that cannot be expressed as simple fractions. These roots are considered irrational because they do not have a terminating or repeating decimal representation. Irrational roots are an important concept in the context of simplifying expressions with roots and solving quadratic equations using the square root property.
Parabola: A parabola is a curved, U-shaped line or surface that is the graph of a quadratic function. It is one of the fundamental conic sections, along with the circle, ellipse, and hyperbola. Parabolas have many important applications in mathematics, physics, and engineering.
Pythagorean Theorem: The Pythagorean Theorem is a fundamental mathematical principle that describes the relationship between the sides of a right triangle. It states that the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
Quadratic Equation: A quadratic equation is a polynomial equation of the second degree, where the highest exponent of the variable is 2. It takes the general form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a$ is not equal to 0. Quadratic equations are fundamental in algebra and have important applications in various fields, including physics, engineering, and economics.
Quadratic Formula: The quadratic formula is a mathematical equation used to solve quadratic equations, which are polynomial equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$. This formula provides a systematic way to find the solutions, or roots, of a quadratic equation.
Radicand: The radicand is the quantity or expression under the radical sign in a radical expression. It represents the value or number that is to be operated on by the radical symbol, such as the square root or cube root.
Real Solutions: Real solutions refer to the solutions or roots of an equation that are real numbers, as opposed to imaginary or complex numbers. This concept is particularly important in the context of solving radical equations, quadratic equations using the square root property, and applications of quadratic equations.
Square Root Property: The square root property is a fundamental concept in algebra that allows for the solution of certain types of quadratic equations. It states that if the square of a number is equal to a given value, then the number itself can be expressed as the positive or negative square root of that value.
Vertex: The vertex of a function or graph is the point where the graph changes direction, either from decreasing to increasing or from increasing to decreasing. It is the turning point of the graph and represents the maximum or minimum value of the function.
Zero Property: The zero property, also known as the additive identity property, is a fundamental concept in mathematics that states that adding or subtracting zero to any number does not change the value of that number. This property is essential in understanding the behavior of real numbers and solving various algebraic equations.
Zero-Product Property: The zero-product property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. This property is crucial in solving quadratic equations using the square root method.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide