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Chapter 2: Problem 33

Find a number \(c\) such that the graph of \(y=x^{2}+6 x+c\) has its vertex on the \(x\) -axis.

Short Answer

Expert verified
The value of \(c\) that makes the vertex of the given quadratic equation lie on the x-axis is \(c = 9\).

Step by step solution

01

Find the x-coordinate of the vertex

Recall that the x-coordinate of the vertex of a quadratic equation of the form \(y=ax^2+bx+c\) can be found using the formula \[x = -\frac{b}{2a}.\] In this case, \(a = 1\) and \(b = 6\). Using the formula, we get \[x = -\frac{6}{2(1)} = -3.\]
02

Find the y-coordinate of the vertex

Plug the x-coordinate of the vertex, \(x = -3\), into the given equation to find the y-coordinate: \[y=(-3)^2+6(-3)+c=9-18+c=-9+c.\] Since the vertex lies on the x-axis, the y-coordinate should be 0. Thus, we have the equation \[0=-9+c.\]
03

Solve for c

Solve the equation from Step 2 to find the value of c: \[0 = -9 + c \Rightarrow c = 9.\] The value of c that makes the vertex of the given quadratic equation lie on the x-axis is \(c = 9\). The quadratic equation that satisfies the given condition is \(y = x^2 + 6x + 9\).

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