 # Essential Question: What are some things the discriminate is used for?

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Essential Question: What are some things the discriminate is used for?

5-8: The Quadratic Formula Sometimes you’ll run into a quadratic expression that cannot be factored. For example:x 2 + 10x + 4 = 0 There is no combination of numbers that multiplies to get 4 and adds to 10. Yet, there are real numbers that exist for x to make that a true statement. There exists a formula that allows you to find the solutions for any quadratic equation, called the QUADRATIC FORMULA

5-8: The Quadratic Formula A quadratic equation written in standard form ax 2 + bx + c = 0 can be solved with the quadratic equation

5-8: The Quadratic Formula The “b 2 – 4ac” underneath the square root is called the discriminant. The discriminant tells us how many (and what type) of solutions we get from the quadratic equation Discriminant

5-8: The Quadratic Equation The Discriminant Determine the type and number of solutions of x 2 + 6x + 8 = 0 a = 1, b = 6, c = 8 Two real solutions Value of discriminantType and Number of solutions b 2 – 4ac > 0 (positive)2 Real Solutions b 2 – 4ac = 0 (zero)1 Real Solution b 2 – 4ac < 0 (negative)0 Real Solutions 2 Imaginary Solutions

5-8: The Quadratic Equation Determine the type and number of solutions x 2 + 6x + 9 = 0 x 2 + 6x + 10 = 0 6 2 – 4(1)(9) = 36 – 36 = 0 1 Real Solution 6 2 – 4(1)(10) = 36 – 40 = -4 2 Imaginary Solutions

5-8: The Quadratic Formula Empirical verification that the formula works x 2 + 8x + 12 = 0can be factored as (x + 6)(x + 2) = 0meaning x = -6 OR x = -2 a = 1, b = 8, c = 12

5-8: The Quadratic Formula Using the quadratic formula to solve a problem that can’t be factored x 2 + 10x + 4 = 0 a = 1, b = 10, c = 4

5-8: The Quadratic Formula Another equation that can’t be factored 2x 2 - 6x + 1 = 0 a = 2, b = -6, c = 1

5-8: The Quadratic Formula Use the quadratic equation to solve 2x 2 = –6x – 7