Recall Simon and his roller coaster design. The shape of the coaster's first hill can be represented using the quadratic equation \( - \frac{1}{2}x^{2} + 10x - 20 = 0 \). Simon wants to determine how long will it take for the roller coaster's train cars to ascend and descend the first hill. To do this, he must start by solving the quadratic equation.
Simon could graph the equation, factor it, or complete the square; but he would rather use the quadratic formula. The quadratic formula gives the solutions of any quadratic equation, once the equation is written in standard form.
Let's Watch
In the video below, the instructor will show how to use the quadratic formula to solve quadratic equations. He will provide a derivation of the quadratic formula from the standard form of the quadratic equation. You can achieve this derivation by completing the square. Then he will work through two examples. Pay close attention to how he ensures each quadratic equation is written in standard form before he begins solving the equation.
You may want to use the study guide to follow along. If so, click the button below to download the study guide.
Hello! There are many techniques for solving quadratic equations: factoring, graphing, and completing the square are some of the most common. However, there is also a generic formula that can be used to solve a quadratic equation in standard form. When using the complete the square technique on the standard form quadratic, the values for x, in terms of A, B, and C are as follows: The opposite of B plus and minus the square root of the quantity B-squared minus four times A times C, all divided by two A will give us our x-values. This is an example of that derivation – read through the process if you are interested (pause).
This is known as the Quadratic Formula, and can be used on any quadratic in standard form, when the A, B and C values are identified.
For example, when rewritten as a standard trinomial equal to zero, the A-value in this problem is two, B is negative one, and C is negative four. Now, by substituting into the formula and simplifying, we see the following: (silent). So, without factoring, graphing, or completing the square, we have the two solutions for the quadratic! Let’s try another. After rewriting in standard form, the A-value in this problem is one, B is four, and C is three. Now, by substituting into the formula and simplifying, we see the following: (silent). Again, without factoring, graphing, or completing the square, we have the two solutions for the quadratic! What was interesting about these solutions is that both are nice, integer answers that could have been obtained by factoring, but the quadratic formula offers an alternative that works in any case.
Is the equation \( - \frac{1}{2}x^{2} + 10x - 20 = 0 \) written in standard form? Explain.
In the equation \( - \frac{1}{2}x^{2} + 10x - 20 = 0 \) what are the values of a, b, and c?
Simon sets up his quadratic formula as \( x = \frac{- (10) \pm \sqrt{{(10)}^{2} - 4\left( \frac{1}{2} \right)( - 20)}}{2\left( \frac{1}{2} \right)} \). Did Simon substitute the values of a, b, and c correctly? Explain.
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Yes, the equation is written in standard form because the structure of the equation matches \( ax^{2} + bx + c = 0 \). |
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The equation is already written in standard form so you can read the values directly from the equation. They are \( a = - \frac{1}{2} \), \( b = 10 \), and \( c = - 20 \). |
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No, the substitution is incorrect because Simon did not include the negative sign on the value of a. |
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