Questions: Consider the following quadratic equation: 15 x^2=16 x+7 Step 2 of 2 : Solve the quadratic equation by factoring. Write your answer in reduced fraction form, if necessary. Answer x=

Consider the following quadratic equation: 15 x^2=16 x+7 Step 2 of 2 : Solve the quadratic equation by factoring. Write your answer in reduced fraction form, if necessary. Answer x=
Transcript text: Consider the following quadratic equation: \[ 15 x^{2}=16 x+7 \] Step 2 of 2 : Solve the quadratic equation by factoring. Write your answer in reduced fraction form, if necessary. Answer How to enter your answer (opens in new window) \[ x= \] $\square$ $\square$
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Solution

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Solution Steps

Step 1: Rearrange the Equation

Rearrange the given quadratic equation into standard form: \[ 15x^2 = 16x + 7 \] Subtract \(16x + 7\) from both sides: \[ 15x^2 - 16x - 7 = 0 \]

Step 2: Factor the Quadratic Equation

To factor the quadratic equation \(15x^2 - 16x - 7 = 0\), we need two numbers that multiply to \(15 \times (-7) = -105\) and add to \(-16\).

The numbers are \(-21\) and \(5\).

Rewrite the middle term using these numbers: \[ 15x^2 - 21x + 5x - 7 = 0 \]

Group the terms: \[ (15x^2 - 21x) + (5x - 7) = 0 \]

Factor by grouping: \[ 3x(5x - 7) + 1(5x - 7) = 0 \]

Factor out the common factor \((5x - 7)\): \[ (3x + 1)(5x - 7) = 0 \]

Step 3: Solve for \(x\)

Set each factor equal to zero and solve for \(x\):

  1. \(3x + 1 = 0\) \[ 3x = -1 \quad \Rightarrow \quad x = -\frac{1}{3} \]

  2. \(5x - 7 = 0\) \[ 5x = 7 \quad \Rightarrow \quad x = \frac{7}{5} \]

Final Answer

The solutions to the quadratic equation are: \[ \boxed{x = -\frac{1}{3}, \frac{7}{5}} \]

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