Use factoring to solve the quadratic equation. Check by substitution or by using a graphing utility and identifying​ x-intercepts.
2 x squared equals 17 x plus 9

The given quadratic equation is:

\[
2x^2 = 17x + 9
\]

### Step 1: Rearrange the equation
To use factoring, we first move all terms to one side of the equation:

\[
2x^2 - 17x - 9 = 0
\]

### Step 2: Factor the quadratic equation
We need to factor \(2x^2 - 17x - 9\). First, find two numbers that multiply to give \(2 \times -9 = -18\) and add up to \(-17\). These numbers are \(-18\) and \(1\).

Now, we rewrite the middle term (\(-17x\)) using these numbers:

\[
2x^2 - 18x + x - 9 = 0
\]

### Step 3: Group the terms and factor by grouping
Group the terms in pairs:

\[
(2x^2 - 18x) + (x - 9) = 0
\]

Now, factor out the greatest common factors from each group:

\[
2x(x - 9) + 1(x - 9) = 0
\]

### Step 4: Factor out the common binomial factor
Now, factor out the common factor \((x - 9)\):

\[
(2x + 1)(x - 9) = 0
\]

### Step 5: Solve for \(x\)
Now set each factor equal to zero and solve for \(x\):

\[
2x + 1 = 0 \quad \text{or} \quad x - 9 = 0
\]

For \(2x + 1 = 0\):

\[
2x = -1 \quad \Rightarrow \quad x = -\frac{1}{2}
\]

For \(x - 9 = 0\):

\[
x = 9
\]

So, the solutions are:

\[
x = -\frac{1}{2} \quad \text{or} \quad x = 9
\]

### Step 6: Check by substitution
Substitute \(x = -\frac{1}{2}\) and \(x = 9\) back into the original equation to verify the solutions.

For \(x = -\frac{1}{2}\):

\[
2\left(-\frac{1}{2}\right)^2 = 17\left(-\frac{1}{2}\right) + 9
\]
\[
2 \times \frac{1}{4} = -\frac{17}{2} + 9
\]
\[
\frac{1}{2} = \frac{-17 + 18}{2} = \frac{1}{2}
\]

This is true, so \(x = -\frac{1}{2}\) is a solution.

For \(x = 9\):

\[
2(9)^2 = 17(9) + 9
\]
\[
2 \times 81 = 153 + 9
\]
\[
162 = 162
\]

This is also true, so \(x = 9\) is a solution.

Both solutions are verified.

### Step 7: Graphing Utility (Optional)
You can verify this graphically by plotting the equation \(2x^2 - 17x - 9 = 0\) and checking the x-intercepts, which should be at \(x = -\frac{1}{2}\) and \(x = 9\).

---

Would you like more details on any step or clarification on the factoring method?

Here are some related questions for further practice:
1. Factor \(3x^2 + 5x - 2 = 0\).
2. Solve \(4x^2 - 12x + 9 = 0\) by factoring.
3. Use the quadratic formula to solve \(x^2 + 7x + 12 = 0\).
4. Verify the solutions of \(x^2 - 5x + 6 = 0\) by graphing.
5. Solve \(6x^2 + 11x - 35 = 0\) by factoring.

**Tip:** When factoring quadratics, always check your work by expanding the factors to ensure they match the original equation.

Use factoring to solve the quadratic equation. Check by substitution or by using a graphing utility and identifying x-intercepts. 2x^2 = 17x + 9.

Learn how to solve the quadratic equation 2x^2 = 17x + 9 by factoring. This detailed guide includes step-by-step factoring, solving for x, and verifying solutions by substitution. Perfect for students in Grades 9-11 studying algebra and quadratic equations.

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