Answer
The quadratic expression \(2x^{2} - 7x + 5\) can be factored as \((2x - 5)(x - 1)\). Solving the equation \(2x^{2} - 7x + 5 = 0\) yields the solutions \(x = 1\) and \(x = \frac{5}{2}\).
Solution
It appears you have the quadratic expression:
\[
2x^{2} - 7x + 5
\]
Assuming you'd like to **factor** this quadratic expression or **solve** the equation \(2x^{2} - 7x + 5 = 0\), here's how you can proceed:
### **1. Factoring the Quadratic Expression**
To factor the quadratic expression \(2x^{2} - 7x + 5\), we look for two numbers that:
- Multiply to \(2 \times 5 = 10\) (the product of the coefficient of \(x^{2}\) and the constant term).
- Add up to \(-7\) (the coefficient of the \(x\) term).
The numbers \(-2\) and \(-5\) satisfy these conditions because:
- \((-2) \times (-5) = 10\)
- \((-2) + (-5) = -7\)
Using these numbers, we can rewrite the middle term and factor by grouping:
\[
2x^{2} - 2x - 5x + 5
\]
Group the terms:
\[
(2x^{2} - 2x) - (5x - 5)
\]
Factor out the common factors in each group:
\[
2x(x - 1) - 5(x - 1)
\]
Now, factor out the common binomial \((x - 1)\):
\[
(2x - 5)(x - 1)
\]
So, the factored form of the quadratic expression is:
\[
2x^{2} - 7x + 5 = (2x - 5)(x - 1)
\]
### **2. Solving the Quadratic Equation**
If you're looking to solve the equation \(2x^{2} - 7x + 5 = 0\), you can use the factored form to find the solutions:
\[
(2x - 5)(x - 1) = 0
\]
Set each factor equal to zero and solve for \(x\):
1. **First Factor:**
\[
2x - 5 = 0 \\
2x = 5 \\
x = \frac{5}{2}
\]
2. **Second Factor:**
\[
x - 1 = 0 \\
x = 1
\]
**Solutions:**
\[
x = 1 \quad \text{and} \quad x = \frac{5}{2}
\]
### **Alternative Method: Quadratic Formula**
If factoring is challenging, you can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}
\]
For the equation \(2x^{2} - 7x + 5 = 0\):
- \(a = 2\)
- \(b = -7\)
- \(c = 5\)
Plugging the values into the formula:
\[
x = \frac{-(-7) \pm \sqrt{(-7)^{2} - 4 \times 2 \times 5}}{2 \times 2} \\
x = \frac{7 \pm \sqrt{49 - 40}}{4} \\
x = \frac{7 \pm \sqrt{9}}{4} \\
x = \frac{7 \pm 3}{4}
\]
This gives two solutions:
1. \[
x = \frac{7 + 3}{4} = \frac{10}{4} = \frac{5}{2}
\]
2. \[
x = \frac{7 - 3}{4} = \frac{4}{4} = 1
\]
**Thus, the solutions are \(x = 1\) and \(x = \frac{5}{2}\).**
Reviewed and approved by the UpStudy tutoring team
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