Question

One solution to a quadratic function, f , is given below: √7 + 51 . Which of the following statements is true about the given function?

A. The other solution to function f is -√7 - 51 .

B. The other solution to function f is -√7 + 51 .

C. Function f has no other solutions.

D. The other solution to function f is √7 - 51 .

Answer 1

The correct statement about the quadratic function f, given one solution as √7 + 51, is that the other solution is √7 - 51. This is due to the properties of quadratic equations where if one solution is real, the second one must also be real and symmetric to the first about the axis of symmetry.

Step-by-step explanation:

One solution to a quadratic function, f, is given as √7 + 51. According to the properties of quadratic equations, if the coefficients are real numbers, the solutions are either both real or complex conjugates. Since √7 + 51 is a real number, the other solution must also be real. By the nature of quadratic equations and their symmetry about the axis of symmetry (which is the line that passes through the vertex and is perpendicular to the x-axis), we can determine that the other solution is √7 - 51. Therefore, the correct statement about function f is:

The other solution to function f is √7 - 51.

To understand why, we can consider the quadratic formula, which for an equation of the form ax² + bx + c = 0 is:

x = (-b ± √(b² - 4ac)) / (2a)

If we have one solution x1 = √7 + 51, the quadratic formula implies that the other solution x2 must involve the same terms but with the opposite sign preceding the square root. Thus, the second solution would be x2 = -√7 +51, which is equivalent to the first solution with the √7 term negated.

Answer 2

The quadratic equation's solutions come in conjugate pairs; therefore, if one solution is √7 + 51, the other is -√7 + 51.Therefore the correct option is B. The other solution to function f is -√7 + 51.

Step-by-step explanation:

The given solution to the quadratic function is √7 + 51. In a quadratic equation of the form ax²+ bx + c = 0, the solutions can be found using the quadratic formula:

`\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \]`

Comparing this with the given solution,
`\( √(7) + 51 \),` we can identify a, b , and c as 1, -51, and -7, respectively. Therefore, the quadratic equation is x² - 51x - 7 = 0. To find the other solution, we use the quadratic formula again with the negative square root:

`\[ x = \frac{{51 - \sqrt{{(-51)^2 - 4(1)(-7)}}}}{{2(1)}} \]`

Simplifying this expression gives us
`\( -√(7) + 51 \)`. Therefore, the correct statement is that the other solution to function f is
`\( -√(7) + 51 \),` making option B the correct choice.

In summary, by using the quadratic formula and substituting the given solution, we determine the correct alternative. The solutions to quadratic equations come in conjugate pairs, so if
`\( √(7) + 51 \)` is one solution,
`\( -√(7) + 51 \)` is the other. This is confirmed by the mathematical calculation, leading to the final answer of option B.

Therefore the correct option is B. The other solution to function f is -√7 + 51.