QUADRATIC EQUATIONS MSJC ~ San Jacinto Campus Math Center Workshop Series Theresa Hert

Radicals with index 2 are referred to as square roots. Simplify Radicals Radicals with index 2 are referred to as square roots.

Simplify Radicals Break down the radicand, the number inside the radical, into prime factors. Circle a pair of matching factors, take out THE factor. Since no operation sign is visible, the “glue” holding everything together is Multiplication. When you bring a factor out of the radical, it gets multiplied to the number in front of the radical.

Simplify the Radical

Simplify Rational Expressions containing Radicals First simplify the radical. To reduce the fraction, Factor. Beware of addition. Plus sign – use one set of parentheses to factor out what is common.

Simplify this Rational Expression containing a Radical

Quadratic Equations contain both an equal sign and a variable with exponent 2. General form: ax2 + bx + c = 0

A quadratic equation is an equation equivalent to an equation of the type ax2 + bx + c = 0, where a is nonzero We can solve a quadratic equation by using the Quadratic Formula

The Quadratic Formula Solve the equation ax2 + bx + c = 0 for x by Completing the Square

The Quadratic Formula Solutions to ax2 + bx + c = 0 for a nonzero are

Solve this Quadratic Equation by using the Quadratic Formula 6y2 – 3y – 5 = 0 a = 6 b = -3 c = -5

because of the addition, 6y2 – 3y – 5 = 0 a = 6 b = -3 c = -5 because of the addition, you can NOT reduce the fraction

Ex: Use the Quadratic Formula to solve x2 + 7x + 6 = 0 1 7 6 Recall: For quadratic equation ax2 + bx + c = 0, the solutions to a quadratic equation are given by Identify a, b, and c in ax2 + bx + c = 0: a = b = c = 1 7 6 Now evaluate the quadratic formula at the identified values of a, b, and c

x2 + 7x + 6 = 0 a = 1 b = 7 c = 6 x = ( - 7 + 5)/2 = - 1 and x = (-7 – 5)/2 = - 6 x = { - 1, - 6 }

Ex: Use the Quadratic Formula to solve 2m2 + m – 10 = 0 2 1 – 10 Recall: For quadratic equation ax2 + bx + c = 0, the solutions to a quadratic equation are given by Identify a, b, and c in am2 + bm + c = 0: a = b = c = 2 1 - 10 Now evaluate the quadratic formula at the identified values of a, b, and c

2x2 + 1x – 10 = 0 a = 2 b = 1 c = -10 m = ( - 1 + 9)/4 = 2 and m = (-1 – 9)/4 = - 5/2 m = { 2, - 5/2 }

Ex: Use the Quadratic Formula to solve x2 + 5x = -3 x2 + 5x + 3 = 0 1 + 5 3 Identify a, b, and c in ax2 + bx + c = 0: a = b = c = 1 + 5 3 Now evaluate the quadratic formula at the identified values of a, b, and c

x2 + 5x + 3 = 0 a = 1 b = 5 c = 3

Ex: Use the Quadratic Formula to solve 10x2 – 5x = 0 10x2 – 5x + 0 = 0 10 - 5 Identify a, b, and c in ax2 + bx + c = 0: a = b = c = 10 - 5 Now evaluate the quadratic formula at the identified values of a, b, and c

10x2 – 5x + 0 = 0 a = 10 b = -5 c = 0

Solve: use the Quadratic Formula.