Sure, let's fill in the first two steps of solving the quadratic equation [tex]\( 2x^2 + x - 10 = 0 \)[/tex] by using factorization.
Step 1: Rewrite the middle term.
We need to split the middle term [tex]\( x \)[/tex] into two terms such that the product of their coefficients equals the product of the coefficient of [tex]\( x^2 \)[/tex] term (which is 2) and the constant term (which is -10).
So, we split [tex]\( x \)[/tex] into [tex]\( -4x \)[/tex] and [tex]\( 5x \)[/tex]:
[tex]\[
2x^2 - 4x + 5x - 10 = 0
\][/tex]
Step 2: Group terms together and find the Greatest Common Factor (GCF).
First, we group the terms:
[tex]\[
(2x^2 - 4x) + (5x - 10) = 0
\][/tex]
Next, we factor out the GCF from each group:
From [tex]\( 2x^2 - 4x \)[/tex], the GCF is [tex]\( 2x \)[/tex]:
[tex]\[
2x(x - 2)
\][/tex]
From [tex]\( 5x - 10 \)[/tex], the GCF is [tex]\( 5 \)[/tex]:
[tex]\[
5(x - 2)
\][/tex]
So, we have:
[tex]\[
2x(x - 2) + 5(x - 2) = 0
\][/tex]
Notice that [tex]\( (x - 2) \)[/tex] is a common factor in both terms, so we can factor this out:
[tex]\[
(2x + 5)(x - 2) = 0
\][/tex]
This completes the first two steps.
Step 3: Apply the Zero Product Property.
According to the Zero Product Property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
2x + 5 = 0 \quad \text{or} \quad x - 2 = 0
\][/tex]
Step 4: Solve.
Solving the first equation, [tex]\( 2x + 5 = 0 \)[/tex]:
[tex]\[
2x = -5
\][/tex]
[tex]\[
x = -\frac{5}{2}
\][/tex]
Solving the second equation, [tex]\( x - 2 = 0 \)[/tex]:
[tex]\[
x = 2
\][/tex]
Thus, the solutions to the equation [tex]\( 2x^2 + x - 10 = 0 \)[/tex] are:
[tex]\[
x = -\frac{5}{2} \quad \text{and} \quad x = 2
\][/tex]
