To solve the quadratic equation [tex]\( 6x^2 - 7x - 24 = 0 \)[/tex], we will use the method of factoring.
Step 1: Identify the coefficients.
The given quadratic equation is [tex]\( 6x^2 - 7x - 24 = 0 \)[/tex].
Here, [tex]\( a = 6 \)[/tex], [tex]\( b = -7 \)[/tex], and [tex]\( c = -24 \)[/tex].
Step 2: Multiply the leading coefficient [tex]\( a \)[/tex] by the constant term [tex]\( c \)[/tex].
Let [tex]\( ac = 6 \times (-24) = -144 \)[/tex].
Step 3: Find two numbers that multiply to [tex]\( -144 \)[/tex] and add up to the middle coefficient [tex]\( b \)[/tex], which is [tex]\( -7 \)[/tex].
The numbers are [tex]\( 9 \)[/tex] and [tex]\( -16 \)[/tex], because:
[tex]\[ 9 \times (-16) = -144 \][/tex]
[tex]\[ 9 + (-16) = -7 \][/tex]
Step 4: Rewrite the middle term [tex]\( -7x \)[/tex] using the two numbers found.
[tex]\[ 6x^2 + 9x - 16x - 24 = 0 \][/tex]
Step 5: Factor by grouping.
Group the terms to facilitate factoring:
[tex]\[ (6x^2 + 9x) + (-16x - 24) = 0 \][/tex]
Factor out the common factor from each group:
[tex]\[ 3x(2x + 3) - 8(2x + 3) = 0 \][/tex]
Step 6: Factor out the common binomial factor [tex]\( 2x + 3 \)[/tex].
[tex]\[ (3x - 8)(2x + 3) = 0 \][/tex]
Step 7: Set each factor equal to zero and solve for [tex]\( x \)[/tex].
[tex]\[ 3x - 8 = 0 \][/tex]
[tex]\[ 3x = 8 \][/tex]
[tex]\[ x = \frac{8}{3} \][/tex]
[tex]\[ 2x + 3 = 0 \][/tex]
[tex]\[ 2x = -3 \][/tex]
[tex]\[ x = -\frac{3}{2} \][/tex]
Thus, the solutions to the equation [tex]\( 6x^2 - 7x - 24 = 0 \)[/tex] are:
[tex]\[ x = \frac{8}{3} \][/tex]
[tex]\[ x = -\frac{3}{2} \][/tex]
