Learn on PengiCalifornia Reveal Math, Algebra 1Unit 10: Quadratic Functions

10-6 Solving Quadratic Equations by Using the Quadratic Formula

In this Grade 9 lesson from California Reveal Math Algebra 1, students learn to solve quadratic equations using the Quadratic Formula, including how to rewrite equations in standard form and identify the coefficients a, b, and c. The lesson also covers the discriminant and how the expression b²−4ac determines the nature of solutions, including irrational roots. Real-world applications, such as modeling blood pressure with a quadratic equation, help students connect the formula to practical problem-solving.

Section 1

Step-by-Step Derivation of the Quadratic Formula

Property

To derive the quadratic formula, start with the general form ax2+bx+c=0ax^2 + bx + c = 0 and complete the square:

  1. Divide by aa: x2+bax+ca=0x^2 + \frac{b}{a}x + \frac{c}{a} = 0
  2. Move the constant: x2+bax=−cax^2 + \frac{b}{a}x = -\frac{c}{a}
  3. Complete the square: x2+bax+b24a2=−ca+b24a2x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} = -\frac{c}{a} + \frac{b^2}{4a^2}
  4. Factor and simplify: (x+b2a)2=b2−4ac4a2(x + \frac{b}{2a})^2 = \frac{b^2 - 4ac}{4a^2}
  5. Extract roots: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Section 2

The quadratic formula

Property

The solutions of the equation

ax2+bx+c=0,a≠0ax^2 + bx + c = 0, \quad a \neq 0

are given by the formula

Section 3

Applying the Quadratic Formula with Decimal and Non-Integer Coefficients

Property

The Quadratic Formula works for any quadratic equation ax2+bx+c=0ax^2 + bx + c = 0, including those with decimal or fractional coefficients. The same formula applies:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Section 4

The Discriminant

Property

In the Quadratic Formula, x=−b±b2−4ac2ax = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}, the quantity b2−4acb^2 - 4ac is called the discriminant.
For a quadratic equation of the form ax2+bx+c=0ax^2 + bx + c = 0, a≠0a \neq 0:

  • If b2−4ac>0b^2 - 4ac > 0, the equation has 2 real solutions.
  • If b2−4ac=0b^2 - 4ac = 0, the equation has 1 real solution.
  • If b2−4ac<0b^2 - 4ac < 0, the equation has no real solutions.

Examples

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Step-by-Step Derivation of the Quadratic Formula

Property

To derive the quadratic formula, start with the general form ax2+bx+c=0ax^2 + bx + c = 0 and complete the square:

  1. Divide by aa: x2+bax+ca=0x^2 + \frac{b}{a}x + \frac{c}{a} = 0
  2. Move the constant: x2+bax=−cax^2 + \frac{b}{a}x = -\frac{c}{a}
  3. Complete the square: x2+bax+b24a2=−ca+b24a2x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} = -\frac{c}{a} + \frac{b^2}{4a^2}
  4. Factor and simplify: (x+b2a)2=b2−4ac4a2(x + \frac{b}{2a})^2 = \frac{b^2 - 4ac}{4a^2}
  5. Extract roots: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Section 2

The quadratic formula

Property

The solutions of the equation

ax2+bx+c=0,a≠0ax^2 + bx + c = 0, \quad a \neq 0

are given by the formula

Section 3

Applying the Quadratic Formula with Decimal and Non-Integer Coefficients

Property

The Quadratic Formula works for any quadratic equation ax2+bx+c=0ax^2 + bx + c = 0, including those with decimal or fractional coefficients. The same formula applies:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Section 4

The Discriminant

Property

In the Quadratic Formula, x=−b±b2−4ac2ax = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}, the quantity b2−4acb^2 - 4ac is called the discriminant.
For a quadratic equation of the form ax2+bx+c=0ax^2 + bx + c = 0, a≠0a \neq 0:

  • If b2−4ac>0b^2 - 4ac > 0, the equation has 2 real solutions.
  • If b2−4ac=0b^2 - 4ac = 0, the equation has 1 real solution.
  • If b2−4ac<0b^2 - 4ac < 0, the equation has no real solutions.

Examples