Question

consider the following quadratic equation:

\\4x^2 - 15x + 1 = -8\\

step 1 of 2: using the standard form \ax^2 + bx + c = 0\\ of the given quadratic equation, factor the left hand side of the equation into two linear factors.

Explanation:

Convert to standard form

Using the Solving Quadratic Equations knowledge point, we first rewrite the equation in standard form \(ax^2 + bx + c = 0\).

$$ 4x^2 - 15x + 1 = -8 $$

Add \(8\) to both sides:

$$ 4x^2 - 15x + 9 = 0 $$

Find factors for grouping

Using the Factoring Trinomials knowledge point, we find two numbers that multiply to \(a \cdot c = 4 \cdot 9 = 36\) and add up to \(b = -15\).
These two numbers are \(-12\) and \(-3\):

$$ (-12) \cdot (-3) = 36 $$

$$ (-12) + (-3) = -15 $$

Rewrite and factor by grouping

Rewrite the middle term using \(-12x\) and \(-3x\):

$$ 4x^2 - 12x - 3x + 9 = 0 $$

Group the terms:

$$ (4x^2 - 12x) - (3x - 9) = 0 $$

Factor out the greatest common factor from each group:

$$ 4x(x - 3) - 3(x - 3) = 0 $$

Factor out the common binomial

Factor out the common binomial factor \((x - 3)\):

$$ (4x - 3)(x - 3) = 0 $$

Answer:

Using the standard form \(ax^2 + bx + c = 0\) of the given quadratic equation, factor the left hand side of the equation into two linear factors.
<blank>\((4x-3)(x-3)\)</blank> \(= 0\)