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Answer :
We start with the equation
[tex]$$
4c^2 - 17c = 15.
$$[/tex]
Step 1. Move all terms to one side
Subtract 15 from both sides of the equation to obtain
[tex]$$
4c^2 - 17c - 15 = 0.
$$[/tex]
Step 2. Factor the quadratic expression
We look for a factorization of the quadratic. We want to express
[tex]$$
4c^2 - 17c - 15
$$[/tex]
in the form
[tex]$$
(c - 5)(4c + 3) = 0.
$$[/tex]
You can verify that expanding [tex]$(c - 5)(4c + 3)$[/tex] reproduces the quadratic:
[tex]$$
(c - 5)(4c + 3) = 4c^2 + 3c - 20c - 15 = 4c^2 - 17c - 15.
$$[/tex]
Step 3. Solve for [tex]\( c \)[/tex]
Set each factor equal to zero:
1. Setting the first factor equal to zero:
[tex]$$
c - 5 = 0 \quad \Longrightarrow \quad c = 5.
$$[/tex]
2. Setting the second factor equal to zero:
[tex]$$
4c + 3 = 0 \quad \Longrightarrow \quad 4c = -3 \quad \Longrightarrow \quad c = -\frac{3}{4}.
$$[/tex]
Final Answer
The solutions to the equation are
[tex]$$
c = 5 \quad \text{or} \quad c = -\frac{3}{4}.
$$[/tex]
[tex]$$
4c^2 - 17c = 15.
$$[/tex]
Step 1. Move all terms to one side
Subtract 15 from both sides of the equation to obtain
[tex]$$
4c^2 - 17c - 15 = 0.
$$[/tex]
Step 2. Factor the quadratic expression
We look for a factorization of the quadratic. We want to express
[tex]$$
4c^2 - 17c - 15
$$[/tex]
in the form
[tex]$$
(c - 5)(4c + 3) = 0.
$$[/tex]
You can verify that expanding [tex]$(c - 5)(4c + 3)$[/tex] reproduces the quadratic:
[tex]$$
(c - 5)(4c + 3) = 4c^2 + 3c - 20c - 15 = 4c^2 - 17c - 15.
$$[/tex]
Step 3. Solve for [tex]\( c \)[/tex]
Set each factor equal to zero:
1. Setting the first factor equal to zero:
[tex]$$
c - 5 = 0 \quad \Longrightarrow \quad c = 5.
$$[/tex]
2. Setting the second factor equal to zero:
[tex]$$
4c + 3 = 0 \quad \Longrightarrow \quad 4c = -3 \quad \Longrightarrow \quad c = -\frac{3}{4}.
$$[/tex]
Final Answer
The solutions to the equation are
[tex]$$
c = 5 \quad \text{or} \quad c = -\frac{3}{4}.
$$[/tex]
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