Use the formula and given information to solve for the indicated variable. Round answers to two decimal places as necessary.
[tex]x=\frac{-b+\sqrt{b^2-4 a c}}{2 a}[/tex]
Given: [tex]a =7, b=6[/tex], and [tex]c =-8[/tex].
Calculate x.
Answer (2)
Substitute the given values a = 7 , b = 6 , and c = − 8 into the formula.
Calculate the discriminant: b 2 − 4 a c = 6 2 − 4 ( 7 ) ( − 8 ) = 260 .
Find the square root of the discriminant: 260 ≈ 16.12 .
Calculate x = 14 − 6 + 16.12 ≈ 0.72 .
Round to two decimal places: 0.72 .
Explanation
Understanding the Problem We are given the formula x = 2 a − b + b 2 − 4 a c and the values a = 7 , b = 6 , and c = − 8 . Our goal is to calculate the value of x .
Substituting the Values First, substitute the given values into the formula: x = 2 × 7 − 6 + 6 2 − 4 × 7 × ( − 8 )
Calculating the Discriminant Next, calculate the value inside the square root: 6 2 − 4 × 7 × ( − 8 ) = 36 + 224 = 260 . So, x = 14 − 6 + 260
Finding the Square Root Now, we find the square root of 260: 260 ≈ 16.12 . Therefore, x = 14 − 6 + 16.12
Calculating x Calculate the value of x: x = 14 10.12 ≈ 0.72
Rounding the Answer Finally, round the answer to two decimal places: x ≈ 0.72 .
Examples
This formula is a variation of the quadratic formula, which is used to find the roots of a quadratic equation. Imagine you are designing a bridge and need to calculate the exact point where a supporting cable should be attached to ensure stability. The quadratic formula, and variations of it like the one used here, can help you find precise measurements for structural integrity.
To find x using the formula x = 2 a − b + b 2 − 4 a c with a = 7 , b = 6 , and c = − 8 , we substitute the values and calculate to get x ≈ 0.72 . This involves calculating the discriminant and finding its square root, followed by substituting back into the equation and simplifying. Rounding gives us the final answer as 0.72 .
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