-4x^2 - 7x = -36
What is the positive solution to the given equation?
Answer (2)
The positive solution to the equation − 4 x 2 − 7 x = − 36 is x = 2.25 after rewriting the equation in standard form and applying the quadratic formula.
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Rewrite the given equation in standard quadratic form: 4 x 2 + 7 x − 36 = 0 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Calculate the two possible solutions: x 1 = 2.25 and x 2 = − 4 .
Identify the positive solution: 2.25 .
Explanation
Rewrite the equation We are given the quadratic equation − 4 x 2 − 7 x = − 36 and asked to find the positive solution. First, we need to rewrite the equation in the standard quadratic form a x 2 + b x + c = 0 .
Standard quadratic form Add 36 to both sides of the equation to get: − 4 x 2 − 7 x + 36 = 0 Multiply both sides by -1 to make the leading coefficient positive: 4 x 2 + 7 x − 36 = 0
Quadratic Formula Now, we can use the quadratic formula to solve for x . The quadratic formula is given by: x = 2 a − b ± b 2 − 4 a c
Apply the formula In our equation, a = 4 , b = 7 , and c = − 36 . Plugging these values into the quadratic formula, we get: x = 2 ( 4 ) − 7 ± 7 2 − 4 ( 4 ) ( − 36 ) x = 8 − 7 ± 49 + 576 x = 8 − 7 ± 625 x = 8 − 7 ± 25
Find the solutions Now we find the two possible solutions for x :
x 1 = 8 − 7 + 25 = 8 18 = 4 9 = 2.25 x 2 = 8 − 7 − 25 = 8 − 32 = − 4
Identify the positive solution We are looking for the positive solution, so we choose x = 4 9 = 2.25 .
Examples
Quadratic equations are useful in many real-world scenarios, such as calculating the trajectory of a ball, determining the dimensions of a garden, or modeling the growth of a population. For example, if you want to build a rectangular garden with an area of 36 square meters and you know that the length must be 7 meters more than 4 times the width, you can use a quadratic equation to find the dimensions of the garden. In this case, the width would be 2.25 meters and the length would be 16 meters.
