Alright, let's solve the quadratic equation [tex]\(2x^2 + 3x - 20 = 0\)[/tex] by factorization step-by-step.
### Step 1: Write the equation in standard form
The equation is already in standard form:
[tex]\[ 2x^2 + 3x - 20 = 0 \][/tex]
### Step 2: Factor the quadratic expression
We need to factor the quadratic expression [tex]\(2x^2 + 3x - 20\)[/tex]. To do this, we look for two numbers that multiply to [tex]\(2 \cdot (-20) = -40\)[/tex] and add up to [tex]\(3\)[/tex].
These numbers are [tex]\(8\)[/tex] and [tex]\(-5\)[/tex].
### Step 3: Split the middle term using the obtained numbers
Rewrite [tex]\(3x\)[/tex] using the numbers [tex]\(8\)[/tex] and [tex]\(-5\)[/tex]:
[tex]\[ 2x^2 + 8x - 5x - 20 \][/tex]
### Step 4: Factor by grouping
Group the terms in pairs:
[tex]\[ (2x^2 + 8x) + (-5x - 20) \][/tex]
Factor out the common factors in each group:
[tex]\[ 2x(x + 4) - 5(x + 4) \][/tex]
Notice that [tex]\((x + 4)\)[/tex] is common in both groups:
[tex]\[ (2x - 5)(x + 4) = 0 \][/tex]
### Step 5: Solve for [tex]\(x\)[/tex] using the Zero Product Property
For the product to be zero, at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[ 2x - 5 = 0 \][/tex]
[tex]\[ 2x = 5 \][/tex]
[tex]\[ x = \frac{5}{2} \][/tex]
[tex]\[ x + 4 = 0 \][/tex]
[tex]\[ x = -4 \][/tex]
### Step 6: State the factorized form and the solutions
The quadratic equation in factorized form is:
[tex]\[ (2x - 5)(x + 4) = 0 \][/tex]
And the solutions of the equation are:
[tex]\[ x = \frac{5}{2} \quad \text{and} \quad x = -4 \][/tex]
So, we have:
- The factorized form: [tex]\((x + 4)(2x - 5)\)[/tex]
- The solutions: [tex]\(x = -4\)[/tex] and [tex]\(x = \frac{5}{2}\)[/tex]
This completes the factorization and solving of the given quadratic equation.
