SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE

SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE, ALGEBRA CAN BE USED TO FIND EXACT SOLUTIONS. ONE OF THOSE ALGEBRAIC METHODS IS “COMPLETING THE SQUARE”

SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE Let’s solve x 2 – 10 x + 18 = 0 Step 1: Get rid of constant on the left side x 2 – 10 x + 18 = 0 -18 x 2 – 10 x -18 = -18

SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE Let’s solve x 2 – 10 x + 18 = 0 Step 2: Add constant to left side to create PST Half of middle term, then square it. Must add it to BOTH sides. x 2 – 10 x + 25 = -18 + 25 (x – 5)2 = 7

SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE Let’s solve x 2 – 10 x + 18 = 0 Step 3: Square root of both sides. (x – 5)2 = 7

SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE Let’s solve x 2 – 10 x + 18 = 0 Step 4: Solve left side for x (x – 5)2 = 7 +5 +5

SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE Try this one x 2 + 6 x – 3 = 0 +3 Half of 6, squared x 2 + 6 x +3 =3 +9 +9 x 2 + 6 x + 9 = 12 (x + 3)2 = 12 -3 -3

COMPLETE THE SQUARE 36 = 3 + _____ 36 x 2 + 12 x + _____

COMPLETE THE SQUARE 16 = 10 + _____ 16 x 2 – 8 x + _____

COMPLETE THE SQUARE 100 = 1 + _____ 100 x 2 – 20 x + _____

SOLVING QUADRATIC EQUATIONS USING THE QUADRATIC FORMULA Standard form for quadratic equations is ax 2 + bx + c = 0 and can be solved using the Quadratic Formula:

SOLVING QUADRATIC EQUATIONS USING THE QUADRATIC FORMULA Example: 3 x 2 + 7 x – 2 = 0

THE DISCRIMINANT In a quadratic formula, the discriminant is the expression under the racical sign. What is the discriminant for 4 x 2 + 2 x – 7 = 0 ? b 2 – 4 ac = 22 – 4(4)(-7) = 4 + 112= 116

THE DISCRIMINANT The discriminant tells you something about the roots of the equation. If the discriminant is negative, (b 2 – 4 ac < 0), then there are no real roots (no solutions). If the discriminant is zero, (b 2 – 4 ac = 0), then there is a double root (one solution). If the discriminant is positive, (b 2 – 4 ac > 0), then there are two real roots.