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« Solving Absolute Value Equations |

Solving Equations: (lesson 3 of 4)

Definition:

A quadratic equation in the variable x is an equation that can be written in the form:

$a{x^2} + bx + c = 0$, $a \ne 0$

where a, b and c represent real number coefficients.

This form is sometimes called the standard form. The term quadratic is used for any equation where the highest power of the variable x is 2. The coefficient a cannot be zero, since otherwise it would be a linear equation.

Formula for solving quadratic equations (known as the **quadratic formula**):

${x_{1,2}} = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$, $a \ne 0$

To solve quadratic equations, substitute the coefficients a, b and c into the quadratic formula.

The expression b^{2} - 4ac shown under the square root sign is called the **discriminant**, because it can "discriminate" between the all possible types of answer:

**type 1:** If b^{2} - 4ac ≥ 0 ⇒ equation has two real roots;

**type 2:** If b^{2} - 4ac = 0 ⇒ equation has two real roots but they are both the same.

**type 3:** If b^{2} - 4ac ≤ 0 ⇒ equation has two complex roots;

Example 1:

Solve the following quadratic equation using the quadratic formula.

2x^{2} + 7x - 15 = 0

Solution:

In this case a = 2 b = 7 c= -15

The value of the discriminant is b^{2} - 4ac = 72 - 4(2)(-15) = 169
(Type 1)

${x_{1,2}} = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} = \frac{{ - 7 \pm \sqrt {169} }}{4} = \frac{{ - 7 \pm 13}}{4} \to {x_1} = - 5$ and ${x_2} = \frac{3}{2}$

Level 1

Level 2

Example 2:

Solve the following equation using the quadratic formula.

4x^{2} - 20x + 25 = 0

Solution:

In this case a = 4 b = - 20 c = 25

The value of the discriminant is b^{2} - 4ac = 202 - 4(4)(25) = 0

${x_{1,2}} = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} = \frac{{20 \pm \sqrt 0 }}{8} = \frac{{20}}{8} = \frac{5}{2} \to x = - \frac{5}{2}$

That is, in this case since the value of the discriminant is zero, the two roots of the equation have the same of 2.5.

Level 1

Level 2

Example 3:

Solve the following quadratic equation using the quadratic formula.

5x^{2} + 2x + 3 = 0

Solution:

In this case a = 5 b = 2 c = 3

The value of the discriminant
is b^{2 }- 4ac = 42 - 4(5)(3) = - 44

Since the value of the discriminant is negative, this equation has no roots that are real numbers.