To factor the quadratic expression \(2u^2 - 9u + 10\), we can use the method of factoring by grouping.
First, we multiply the coefficient of \(u^2\) (which is 2) by the constant term (which is 10): \[ 2 \times 10 = 20 \] Next, we look for two numbers that multiply to 20 and add to -9. The numbers that meet these criteria are -4 and -5, since: \[ -4 \times -5 = 20 \quad \text{and} \quad -4 + -5 = -9 \] Now, we can rewrite the middle term \(-9u\) using -4 and -5: \[ 2u^2 - 4u - 5u + 10 \] Next, we group the terms: \[ (2u^2 - 4u) + (-5u + 10) \] Now we factor out the common factors from each group: \[ 2u(u - 2) - 5(u - 2) \] Now we can factor out the common binomial factor \((u - 2)\): \[ (2u - 5)(u - 2) \]
So, the completely factored form of the expression \(2u^2 - 9u + 10\) is: \[ (2u - 5)(u - 2) \]