Learn on PengiYoshiwara Elementary AlgebraChapter 6: Quadratic Equations

Lesson 5: The Quadratic Formula

In this Grade 6 lesson from Yoshiwara Elementary Algebra (Chapter 6: Quadratic Equations), students learn to apply the quadratic formula x = (−b ± √(b²−4ac)) / 2a to solve any quadratic equation in standard form, including cases where factoring is not possible. The lesson covers identifying coefficients a, b, and c, clearing fractional coefficients using the LCD before substituting, and using a calculator to find decimal approximations of exact radical solutions.

Section 1

📘 The Quadratic Formula

New Concept

The quadratic formula is your master key for solving any quadratic equation, ax2+bx+c=0ax^2 + bx + c = 0. It finds the exact solutions—even when factoring doesn't work—and tells us the precise x-intercepts of the equation's graph.

What’s next

Now that you have the formula, you'll apply it in interactive examples and practice problems to find exact solutions and x-intercepts of parabolas.

Section 2

The quadratic formula

Property

The solutions of the equation

ax2+bx+c=0,a0ax^2 + bx + c = 0, \quad a \neq 0

are given by the formula

Section 3

Standard form for quadratic formula

Property

Before we use the quadratic formula, we must write the equation in standard form, ax2+bx+c=0ax^2 + bx + c = 0, so that we can identify the coefficients aa, bb, and cc. If the coefficients are fractions, it helps to clear the fractions by multiplying each term by the LCD.

Examples

  • To solve 5x2=8x35x^2 = 8x - 3, first rearrange it to standard form: 5x28x+3=05x^2 - 8x + 3 = 0. Now you can correctly identify a=5,b=8,a=5, b=-8, and c=3c=3.
  • Given the equation x2x3=43x^2 - \frac{x}{3} = \frac{4}{3}, first multiply by the LCD, 3, to get 3x2x=43x^2 - x = 4. Then, rewrite in standard form as 3x2x4=03x^2 - x - 4 = 0, so a=3,b=1,c=4a=3, b=-1, c=-4.

Section 4

Non-real solutions

Property

If the value under the radical in the quadratic formula, b24acb^2 - 4ac, is a negative number, the equation has no real-valued solutions. This is because the square root of a negative number is not a real number. The solutions are a type of number called complex numbers.

Examples

  • For x24x+5=0x^2 - 4x + 5 = 0, the formula gives x=4±(4)24(1)(5)2=4±16202=4±42x = \frac{4 \pm \sqrt{(-4)^2 - 4(1)(5)}}{2} = \frac{4 \pm \sqrt{16-20}}{2} = \frac{4 \pm \sqrt{-4}}{2}. Since 4\sqrt{-4} is not a real number, there are no real solutions.
  • To solve x2+25=0x^2 + 25 = 0, the formula with a=1,b=0,c=25a=1, b=0, c=25 gives x=0±024(1)(25)2=±1002x = \frac{0 \pm \sqrt{0^2 - 4(1)(25)}}{2} = \frac{\pm \sqrt{-100}}{2}. There are no real solutions.

Section 5

Number of x-intercepts

Property

The xx-intercepts of the graph of y=ax2+bx+cy = ax^2 + bx + c are the solutions of ax2+bx+c=0ax^2 + bx + c = 0. There are three possibilities:

  1. If both solutions are real numbers, and unequal, the graph has two xx-intercepts.
  1. If the solutions are real and equal, the graph has one xx-intercept, which is also its vertex.

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 6: Quadratic Equations

  1. Lesson 1

    Lesson 1: Extracting Roots

  2. Lesson 2

    Lesson 2: Some Quadratic Models

  3. Lesson 3

    Lesson 3: Solving Quadratic Equations by Factoring

  4. Lesson 4

    Lesson 4: Graphing Quadratic Equations

  5. Lesson 5Current

    Lesson 5: The Quadratic Formula

  6. Lesson 6

    Lesson 6: Chapter Summary and Review

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

📘 The Quadratic Formula

New Concept

The quadratic formula is your master key for solving any quadratic equation, ax2+bx+c=0ax^2 + bx + c = 0. It finds the exact solutions—even when factoring doesn't work—and tells us the precise x-intercepts of the equation's graph.

What’s next

Now that you have the formula, you'll apply it in interactive examples and practice problems to find exact solutions and x-intercepts of parabolas.

Section 2

The quadratic formula

Property

The solutions of the equation

ax2+bx+c=0,a0ax^2 + bx + c = 0, \quad a \neq 0

are given by the formula

Section 3

Standard form for quadratic formula

Property

Before we use the quadratic formula, we must write the equation in standard form, ax2+bx+c=0ax^2 + bx + c = 0, so that we can identify the coefficients aa, bb, and cc. If the coefficients are fractions, it helps to clear the fractions by multiplying each term by the LCD.

Examples

  • To solve 5x2=8x35x^2 = 8x - 3, first rearrange it to standard form: 5x28x+3=05x^2 - 8x + 3 = 0. Now you can correctly identify a=5,b=8,a=5, b=-8, and c=3c=3.
  • Given the equation x2x3=43x^2 - \frac{x}{3} = \frac{4}{3}, first multiply by the LCD, 3, to get 3x2x=43x^2 - x = 4. Then, rewrite in standard form as 3x2x4=03x^2 - x - 4 = 0, so a=3,b=1,c=4a=3, b=-1, c=-4.

Section 4

Non-real solutions

Property

If the value under the radical in the quadratic formula, b24acb^2 - 4ac, is a negative number, the equation has no real-valued solutions. This is because the square root of a negative number is not a real number. The solutions are a type of number called complex numbers.

Examples

  • For x24x+5=0x^2 - 4x + 5 = 0, the formula gives x=4±(4)24(1)(5)2=4±16202=4±42x = \frac{4 \pm \sqrt{(-4)^2 - 4(1)(5)}}{2} = \frac{4 \pm \sqrt{16-20}}{2} = \frac{4 \pm \sqrt{-4}}{2}. Since 4\sqrt{-4} is not a real number, there are no real solutions.
  • To solve x2+25=0x^2 + 25 = 0, the formula with a=1,b=0,c=25a=1, b=0, c=25 gives x=0±024(1)(25)2=±1002x = \frac{0 \pm \sqrt{0^2 - 4(1)(25)}}{2} = \frac{\pm \sqrt{-100}}{2}. There are no real solutions.

Section 5

Number of x-intercepts

Property

The xx-intercepts of the graph of y=ax2+bx+cy = ax^2 + bx + c are the solutions of ax2+bx+c=0ax^2 + bx + c = 0. There are three possibilities:

  1. If both solutions are real numbers, and unequal, the graph has two xx-intercepts.
  1. If the solutions are real and equal, the graph has one xx-intercept, which is also its vertex.

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 6: Quadratic Equations

  1. Lesson 1

    Lesson 1: Extracting Roots

  2. Lesson 2

    Lesson 2: Some Quadratic Models

  3. Lesson 3

    Lesson 3: Solving Quadratic Equations by Factoring

  4. Lesson 4

    Lesson 4: Graphing Quadratic Equations

  5. Lesson 5Current

    Lesson 5: The Quadratic Formula

  6. Lesson 6

    Lesson 6: Chapter Summary and Review