Use factoring to solve the quadratic equation. Check by substitution or by using a graphing utility and identifying​ x-intercepts.
5 x squared equals 23 x plus 42
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Part 1
Rewrite the equation in factored form.
  
left parenthesis negative seven fifths comma 6 right parenthesis
equals0
​(Factor completely.)

To solve the quadratic equation \(5x^2 = 23x + 42\) by factoring, we will first rewrite the equation in standard form, which is \(ax^2 + bx + c = 0\).

### Step 1: Rewrite the equation in standard form

Move all terms to one side of the equation:
\[
5x^2 - 23x - 42 = 0
\]

### Step 2: Factor the quadratic equation

We need to factor the quadratic equation \(5x^2 - 23x - 42 = 0\). To factor it, we look for two numbers that multiply to \(5 \times (-42) = -210\) and add to \(-23\).

These numbers are \(-30\) and \(7\), since:
\[
-30 \times 7 = -210 \quad \text{and} \quad -30 + 7 = -23
\]

So, we can rewrite the middle term \(-23x\) using these numbers:
\[
5x^2 - 30x + 7x - 42 = 0
\]

Next, factor by grouping:
\[
(5x^2 - 30x) + (7x - 42) = 0
\]
\[
5x(x - 6) + 7(x - 6) = 0
\]

Now, factor out the common binomial factor \((x - 6)\):
\[
(5x + 7)(x - 6) = 0
\]

### Step 3: Solve for \(x\)

Set each factor equal to zero:
\[
5x + 7 = 0 \quad \text{or} \quad x - 6 = 0
\]

Solve each equation:
\[
5x + 7 = 0 \quad \Rightarrow \quad 5x = -7 \quad \Rightarrow \quad x = -\frac{7}{5}
\]
\[
x - 6 = 0 \quad \Rightarrow \quad x = 6
\]

So, the solutions are \(x = -\frac{7}{5}\) and \(x = 6\).

### Step 4: Check the solutions by substitution

Substitute \(x = -\frac{7}{5}\) and \(x = 6\) back into the original equation to verify.

For \(x = -\frac{7}{5}\):
\[
5\left(-\frac{7}{5}\right)^2 = 23\left(-\frac{7}{5}\right) + 42
\]
\[
5\left(\frac{49}{25}\right) = -\frac{161}{5} + 42
\]
\[
\frac{245}{25} = \frac{245}{25} \quad \text{(True)}
\]

For \(x = 6\):
\[
5(6)^2 = 23(6) + 42
\]
\[
5(36) = 138 + 42
\]
\[
180 = 180 \quad \text{(True)}
\]

Both solutions are correct.

### Step 5: Check by using a graphing utility
Using a graphing calculator or software, graph the equation \(5x^2 - 23x - 42 = 0\). The \(x\)-intercepts should occur at \(x = -\frac{7}{5}\) and \(x = 6\), which confirms the solutions.

---

Would you like any further details or have any questions?

Here are 5 related questions:
1. How do you determine when a quadratic equation is factorable?
2. What is the importance of the discriminant in solving quadratic equations?
3. Can every quadratic equation be factored?
4. How do you graph quadratic functions and identify their key features?
5. What are some common mistakes when solving quadratic equations by factoring?

**Tip:** Always check your factored solutions by substituting them back into the original equation to verify their accuracy.

Use factoring to solve the quadratic equation. Check by substitution or by using a graphing utility and identifying​ x-intercepts. 5x squared equals 23x plus 42.

Learn how to solve the quadratic equation 5x^2 = 23x + 42 using factoring. This step-by-step solution covers factoring, substitution, and graphing techniques to find the x-intercepts. Suitable for students in Grades 9-10.

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