Solve the following quadratic equation by factoring using the perfect square trinomial pattern: 4x2+52x+169=0

.(1 point)
Responses

x=6.5
x equals 6.5

x=−8.45
x equals negative 8.45

x=−6.5
x equals negative 6.5

x=8.45

1 answer

To solve the quadratic equation \(4x^2 + 52x + 169 = 0\) by factoring, we need to check if it can be expressed as a perfect square trinomial. A perfect square trinomial has the form \((ax + b)^2 = a^2x^2 + 2abx + b^2\).

Let's start by rewriting the equation:

  1. Factor out the coefficient of \(x^2\) (which is 4 in this case):

    \[ 4(x^2 + 13x) + 169 = 0 \]

  2. Now, rearrange the equation:

    \[ 4(x^2 + 13x + \text{something}) + 169 - \text{something} = 0 \]

    We can complete the square for the expression inside the parentheses. To do this, we take half of the coefficient of \(x\) (which is 13), square it, and add it inside:

    Half of 13 is \( \frac{13}{2} \) and squaring it gives:

    \[ \left(\frac{13}{2}\right)^2 = \frac{169}{4} \]

    Now we can express this trinomial as a perfect square:

    \[ x^2 + 13x + \frac{169}{4} = \left(x + \frac{13}{2}\right)^2 \]

    Therefore, we rewrite our original equation after completing the square:

    \[ 4\left(x + \frac{13}{2}\right)^2 - 4 \cdot \frac{169}{4} + 169 = 0 \]

    Simplifying gives:

    \[ 4\left(x + \frac{13}{2}\right)^2 = 0 \]

  3. Set the equation to zero:

    \[ \left(x + \frac{13}{2}\right)^2 = 0 \]

    Taking the square root of both sides, we find:

    \[ x + \frac{13}{2} = 0 \]

    So,

    \[ x = -\frac{13}{2} = -6.5 \]

The solution to the quadratic equation \(4x^2 + 52x + 169 = 0\) is \(x = -6.5\).

Answer: \(x = -6.5\)