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To determine which values of [tex]\(x\)[/tex] are solutions to the equation [tex]\(4x^2 - 7x = 3x + 24\)[/tex], we will follow these steps:
### Step 1: Simplify the Equation
First, let's isolate all the terms on one side of the equation to form a standard quadratic equation. Starting with:
[tex]\[ 4x^2 - 7x = 3x + 24 \][/tex]
Subtract [tex]\(3x\)[/tex] and 24 from both sides to get:
[tex]\[ 4x^2 - 7x - 3x - 24 = 0 \][/tex]
Combine like terms:
[tex]\[ 4x^2 - 10x - 24 = 0 \][/tex]
### Step 2: Solve the Quadratic Equation
Next, we need to solve the quadratic equation [tex]\( 4x^2 - 10x - 24 = 0 \)[/tex] to find the solutions for [tex]\(x\)[/tex]. This can be done via factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 4 \)[/tex], [tex]\( b = -10 \)[/tex], and [tex]\( c = -24 \)[/tex].
Calculate the discriminant:
[tex]\[ b^2 - 4ac = (-10)^2 - 4 \cdot 4 \cdot (-24) = 100 + 384 = 484 \][/tex]
Find the square root of the discriminant:
[tex]\[ \sqrt{484} = 22 \][/tex]
Now apply the quadratic formula:
[tex]\[ x = \frac{-(-10) \pm 22}{2 \cdot 4} \][/tex]
[tex]\[ x = \frac{10 \pm 22}{8} \][/tex]
This gives us two potential solutions:
[tex]\[ x = \frac{32}{8} = 4 \][/tex]
[tex]\[ x = \frac{-12}{8} = -\frac{3}{2} \][/tex]
So, the solutions to the quadratic equation [tex]\( 4x^2 - 10x - 24 = 0 \)[/tex] are:
[tex]\[ x = 4 \quad \text{and} \quad x = -\frac{3}{2} \][/tex]
### Step 3: Verify the Given Options
We need to check if these solutions match any provided options:
- [tex]\( x = -4 \)[/tex]
- [tex]\( x = -3 \)[/tex]
- [tex]\( x = -\frac{3}{2} \)[/tex]
- [tex]\( x = \frac{2}{3} \)[/tex]
- [tex]\( x = 2 \)[/tex]
- [tex]\( x = 4 \)[/tex]
From our calculation, the solutions to the equation are:
[tex]\[ x = -\frac{3}{2} \quad \text{( equivalent to } -1.5\text{)} \][/tex]
[tex]\[ x = 4 \][/tex]
Checking against the given options, the correct solutions are:
[tex]\[ x = -\frac{3}{2} \quad \text{and} \quad x = 4 \][/tex]
Thus, out of the given options, the values of [tex]\(x\)[/tex] that satisfy the equation [tex]\(4 x^2 - 10 x - 24 = 0\)[/tex] are:
[tex]\[ x = -\frac{3}{2} \quad \text{and} \quad x = 4 \][/tex]
### Step 1: Simplify the Equation
First, let's isolate all the terms on one side of the equation to form a standard quadratic equation. Starting with:
[tex]\[ 4x^2 - 7x = 3x + 24 \][/tex]
Subtract [tex]\(3x\)[/tex] and 24 from both sides to get:
[tex]\[ 4x^2 - 7x - 3x - 24 = 0 \][/tex]
Combine like terms:
[tex]\[ 4x^2 - 10x - 24 = 0 \][/tex]
### Step 2: Solve the Quadratic Equation
Next, we need to solve the quadratic equation [tex]\( 4x^2 - 10x - 24 = 0 \)[/tex] to find the solutions for [tex]\(x\)[/tex]. This can be done via factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 4 \)[/tex], [tex]\( b = -10 \)[/tex], and [tex]\( c = -24 \)[/tex].
Calculate the discriminant:
[tex]\[ b^2 - 4ac = (-10)^2 - 4 \cdot 4 \cdot (-24) = 100 + 384 = 484 \][/tex]
Find the square root of the discriminant:
[tex]\[ \sqrt{484} = 22 \][/tex]
Now apply the quadratic formula:
[tex]\[ x = \frac{-(-10) \pm 22}{2 \cdot 4} \][/tex]
[tex]\[ x = \frac{10 \pm 22}{8} \][/tex]
This gives us two potential solutions:
[tex]\[ x = \frac{32}{8} = 4 \][/tex]
[tex]\[ x = \frac{-12}{8} = -\frac{3}{2} \][/tex]
So, the solutions to the quadratic equation [tex]\( 4x^2 - 10x - 24 = 0 \)[/tex] are:
[tex]\[ x = 4 \quad \text{and} \quad x = -\frac{3}{2} \][/tex]
### Step 3: Verify the Given Options
We need to check if these solutions match any provided options:
- [tex]\( x = -4 \)[/tex]
- [tex]\( x = -3 \)[/tex]
- [tex]\( x = -\frac{3}{2} \)[/tex]
- [tex]\( x = \frac{2}{3} \)[/tex]
- [tex]\( x = 2 \)[/tex]
- [tex]\( x = 4 \)[/tex]
From our calculation, the solutions to the equation are:
[tex]\[ x = -\frac{3}{2} \quad \text{( equivalent to } -1.5\text{)} \][/tex]
[tex]\[ x = 4 \][/tex]
Checking against the given options, the correct solutions are:
[tex]\[ x = -\frac{3}{2} \quad \text{and} \quad x = 4 \][/tex]
Thus, out of the given options, the values of [tex]\(x\)[/tex] that satisfy the equation [tex]\(4 x^2 - 10 x - 24 = 0\)[/tex] are:
[tex]\[ x = -\frac{3}{2} \quad \text{and} \quad x = 4 \][/tex]
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