1. Chapter 4 Quadratic Equations

2. 4.1 Graphical Solutions of Quadratic Equations
You can solve a quadratic equation of the form ax2 + bx + c = 0 by graphing the corresponding quadratic function, f(x) = ax2 + bx + c. The solutions to a quadratic equation are called the roots of the equation.
You can find the roots by determining the x-intercepts of the graph, or the zeros of the quadratic function. The zeros of a function are the values of x for which f(x) = 0.
The graph of a quadratic function can have zero, one, or two real x-intercepts. Therefore, a quadratic equation has zero, one, or two real roots.

3. 4.2 Factoring Quadratic Equations
Another way of solving a quadratic equation is by factoring.
To solve the equation:
first write the equation in the form of ax2 + bx + c = 0
then factor the left side
Next, set each factor to zero and solve for the unknown
Ex. x2 + 8x = -12
x2 + 8x + 12 = 0
(x + 6)(x + 2) = 0
x + 2 = 0 or x + 6 = 0
x = or x = - 6
Check both answers by substituting the x value:
(-2)2 + 8(-2) = (-6)2 + 8(-6) = -12
= = -12
-12 = - 12 ✓ = -12 ✓

4. 4.3 Solving the Quadratic Equation by Completing the Square
As recalled, completing the square is the process of rewriting a quadratic polynomial from the standard form ax2 + bx + c, to the vertex form, a(x - p)2 + q.
You can use this method to determine the roots of a quadratic equation.
Ex. 2x2 - 4x - 2 = 0
x2 - 2x - 1 = 0 Divide both sides by a common factor of 2
x2 - 2x = 1 Isolate the variable terms on the left side
x2 - 2x + 1 = Complete the square on both sides
(x - 1)2 = 2
√(x - 1)2 = √ 2 Take the square root of both sides
x - 1 = ±√2
x = ±√2 + 1
+√2 + 1 ≈ √2 + 1 ≈ -0.41
x ≈ x ≈ -0.41

5. 4.4 The Quadratic Formula
An alternate method to solve a quadratic equation is using the quadratic formula in the form of
ax2 + bx + c = 0.
The formula is:
x = -b2 ± √ b2 - 4ac 2a
There is also the discriminant which is the expression b2 - 4ac located under the radical sign in the formula. You can use the discriminant to determine the nature of the roots for a quadratic equation.
when the value of the discriminant is positive, b2 - 4ac > 0, there are two distinct roots
when the value of the discriminant is zero, b2 - 4ac = 0, there is one distinct root
when the value of the discriminant is negative, b2 - 4ac < 0, there are no real roots

6. Example: Use the quadratic formula to solve 6x2 - 14x + 8 = 0
x = -b2 ± √ b2 - 4ac 2a
x = -(-14) ± √ (-14)2 - 4(6)(8)
2(8)
x = 14 ± √ 4 16
x = 14 ± 2
x = or x =
x = x = 12
x = 1 x = 3 4