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QUADRATIC EQUATIONS •A quadratic equation is an equation equivalent to one of the form Section 1.2 2 0 where , , and are real numbers and 0. Quadratic Equations • The form above is called the standard form of a quadratic equation. • A quadratic equation is also called a second‐ degree equation.

SOLVING QUADRATIC SOLVING A QUADRATIC EQUATIONS EQUATION BY FACTORING

Quadratic equations can be solved by several 1. Write the quadratic equation in standard methods. form. • By factoring and using the Zero‐Product 2. Factor the quadratic expression. Property 3. Use the Zero‐Product Property to set each • By taking square roots factor equal to 0. • By completing the square 4. Solve the linear equations resulting from Step 3. • By using the Quadratic Formula 5. Check your solution(s).

SOLVING BY THE SQUARE REPEATED SOLUTIONS ROOT METHOD When the left side factors into two linear 1. Write the quadratic in the form of 2 , equations with the same solution, the where is a linear expression and is a quadratic equation is said to have a repeated real number. solution. We also call this solution a root of 2. Take the square root of both sides. multiplicity 2, or a double root. 3. Solve the resulting linear equations if necessary. 4. Check your solution(s).

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SQUARES OF A BINOMIAL COMPLETING THE SQUARE

Square of a Binomial Perfect‐Square Trinomial Adding, to a binomial of the form 2 , the constant that makes the binomial a perfect 5 2 1025 square trinomial is called completing the square. 42 816 To complete the square, In each perfect-square trinomial, the 1. Divide the coefficient of by 2; that is, . coefficient of is 1, and the constant term of 2. Square the result of Step 1. the perfect-square trinomial is the square of 3. Add the result of Step 2. half the coefficient of its term. The resulting trinomial is a perfect square.

SOLVING A QUADRATIC EQUATION SOLVING A QUADRATIC EQUATION BY COMPLETING THE SQUARE BY THE QUADRATIC FORMULA 1. Write the quadratic equation in the form Consider the equation 2 . 2 0, 0 2. Factor the coefficient of 2 out of the left side of the equation. If 2 4 0, this equation has no real solution. 3. Complete the square of the resulting If 2 4 0, the real solution(s) of this binomial. Be sure to maintain equality by equation is (are) given by the quadratic formula. adding the same quantity to both sides. 4. Solve the equation resulting from Step 3 by 4 the Square Root Method. 2 5. Check your solution(s).

THE DISCRIMINANT USING THE DISCRIMINANT

2 In the Quadratic Formula, the quantity For a quadratic equation 0: is called the discriminant. 1. If 2 4 0, there are two unequal This discriminant can tell us if the equation has real solutions. real solutions and how many to expect. 2. If 4 0, there is a repeated real solution, a root of multiplicity 2. 3. If 2 4 0, there is no real solution.

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