Quadratic Formula

A quadratic equation is an equation that contains a squared variable as its highest power on any variable. The general form of a quadratic equation is:

ax² + bx + c = 0

Where a, b, and c are constants and a ≠ 0. In other words there must be a x² term.

Some examples are:

x² + 3x - 3 = 0
4x² + 9 = 0 (Where b = 0)
x² + 5x = 0 (where c = 0)

One way to solve a quadratic equation is by factoring the trinomial. But this only works well for equations that can be factored easily. See PART I and PART II of Factoring Quadratics.

Another method is called completing the square. See PART I and PART II of Completing the Square. From this method a formula was derived to solve directly for x.

A third method for solving quadratic equations uses the derived formula from the completing the square method, called the quadratic formula.

This discussion is focused on solving quadratic equations using the quadratic formula.

Quadratic Formula

x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

Let's solve the following equation using the quadratic formula:

7x² - 17x + 9 = 0

Step 1: Write the equation in the general form ax² + bx + c = 0.

This equation is already in the proper form where a = 7, b = -17 and c = 9.

7x² - 17x + 9 = 0

Step 2: Substitute the values of a, b and c into the quadratic equation.

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

$x = \frac{- (- 17) \pm \sqrt{{(- 17)}^{2} - 4 (7) (9)}}{2 (7)}$

Step 3: Solve for x.

When simplifying the fraction remember that both terms in the numerator must be divisible by the common factor.

In this case $17, \sqrt{541} a n d 14$ do not have a common factor and thus the answer is simplified.

$x = \frac{17 \pm \sqrt{289 - 252}}{14}$

$x = \frac{17 \pm \sqrt{541}}{14}$

Example 1: 16x² = 12x + 4

Step 1: Write the equation in the general form ax² + bx + c = 0.

Where a = 16, b = -12 and c = 4.

16x² - 12x - 4 = 0

Step 2: Substitute the values of a, b and c into the quadratic equation.

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

$x = \frac{- (- 12) \pm \sqrt{{(- 12)}^{2} - 4 (16) (- 4)}}{2 (16)}$

Step 3: Solve for x.

When simplifying the fraction remember that both terms in the numerator must be divisible by the common factor.

In this case 12, 20, and 32 are all divisible by 4 and thus the fraction can be simplified by dividing by $\frac{4}{4}$ .

$x = \frac{12 \pm \sqrt{144 + 256}}{32}$

$x = \frac{12 \pm \sqrt{400}}{32}$

$x = \frac{12 \pm 20}{32}$

$x = \frac{3 \pm 5}{8}$

$x = \frac{8}{8} = 1$ , or $x = \frac{- 2}{8} = \frac{- 1}{4}$

Example 2: 5x + 21x² = 3

Step 1: Write the equation in the general form ax² + bx + c = 0.

Where a = 21, b = 5 and c = -3.

21x² + 5x - 3 = 0

Step 2: Substitute the values of a, b and c into the quadratic equation.

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

$x = \frac{- (5) \pm \sqrt{{(5)}^{2} - 4 (21) (- 3)}}{2 (21)}$

Step 3: Solve for x.

When simplifying the fraction remember that both terms in the numerator must be divisible by the common factor.

In this case $- 5, \sqrt{277} a n d 42$ do not have a common factor and thus the answer is simplified.

$x = \frac{- 5 \pm \sqrt{25 + 252}}{42}$

$x = \frac{- 5 \pm \sqrt{277}}{42}$

In this discussion we encounter cases where the value under the radical sign, called the discriminant, was positive.

If the sign of this value is negative the answer will contain an imaginary number.

Related Links:
Math
algebra
The Difference of Perfect Squares
Introduction and Simple Equations
Algebra Topics

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