Math / Algebra

QuestionFactor the following expression completely: $6 x^{2}+5 x-4=$

Studdy Solution

STEP 1

1. The expression is a quadratic polynomial.
2. We need to factor it into a product of two binomials.

STEP 2

1. Identify the form of the quadratic expression.
2. Use the AC method to factor the quadratic expression.
3. Split the middle term based on the AC method.
4. Factor by grouping.

STEP 3

Identify the form of the quadratic expression. The expression is in the standard form $ax^2 + bx + c$ , where $a = 6$ , $b = 5$ , and $c = -4$ .

STEP 4

Use the AC method to factor the quadratic expression. Calculate the product $ac = 6 \times (-4) = -24$ .

STEP 5

Find two numbers that multiply to $-24$ and add up to $5$ . These numbers are $8$ and $-3$ .
$8 \times (-3) = -24$ $8 + (-3) = 5$

STEP 6

Split the middle term $5x$ using the numbers found:
$6x^2 + 8x - 3x - 4$

STEP 7

Factor by grouping. Group the terms:
$(6x^2 + 8x) + (-3x - 4)$
Factor out the greatest common factor from each group:
$2x(3x + 4) - 1(3x + 4)$
Notice that $(3x + 4)$ is a common factor:
$(2x - 1)(3x + 4)$
The completely factored form of the expression is:
$\boxed{(2x - 1)(3x + 4)}$

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QuestionFactor the following expression completely: $6 x^{2}+5 x-4=$

Studdy Solution

STEP 1

1. The expression is a quadratic polynomial.
2. We need to factor it into a product of two binomials.

STEP 2

1. Identify the form of the quadratic expression.
2. Use the AC method to factor the quadratic expression.
3. Split the middle term based on the AC method.
4. Factor by grouping.

STEP 3

Identify the form of the quadratic expression. The expression is in the standard form $ax^2 + bx + c$ , where $a = 6$ , $b = 5$ , and $c = -4$ .

STEP 4

Use the AC method to factor the quadratic expression. Calculate the product $ac = 6 \times (-4) = -24$ .

STEP 5

Find two numbers that multiply to $-24$ and add up to $5$ . These numbers are $8$ and $-3$ .
$8 \times (-3) = -24$ $8 + (-3) = 5$

STEP 6

Split the middle term $5x$ using the numbers found:
$6x^2 + 8x - 3x - 4$

STEP 7

Factor by grouping. Group the terms:
$(6x^2 + 8x) + (-3x - 4)$
Factor out the greatest common factor from each group:
$2x(3x + 4) - 1(3x + 4)$
Notice that $(3x + 4)$ is a common factor:
$(2x - 1)(3x + 4)$
The completely factored form of the expression is:
$\boxed{(2x - 1)(3x + 4)}$

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QuestionFactor the following expression completely: 6x2+5x−4=6 x^{2}+5 x-4=6x2+5x−4=

Studdy Solution

STEP 1

1. The expression is a quadratic polynomial.2. We need to factor it into a product of two binomials.

STEP 2

1. Identify the form of the quadratic expression.2. Use the AC method to factor the quadratic expression.3. Split the middle term based on the AC method.4. Factor by grouping.

STEP 3

Identify the form of the quadratic expression. The expression is in the standard form ax2+bx+c ax^2 + bx + c ax2+bx+c, where a=6 a = 6 a=6, b=5 b = 5 b=5, and c=−4 c = -4 c=−4.

STEP 4

Use the AC method to factor the quadratic expression. Calculate the product ac=6×(−4)=−24 ac = 6 \times (-4) = -24 ac=6×(−4)=−24.

STEP 5

Find two numbers that multiply to −24 -24 −24 and add up to 5 5 5. These numbers are 8 8 8 and −3 -3 −3.8×(−3)=−24 8 \times (-3) = -24 8×(−3)=−24 8+(−3)=5 8 + (-3) = 5 8+(−3)=5

STEP 6

Split the middle term 5x 5x 5x using the numbers found:6x2+8x−3x−4 6x^2 + 8x - 3x - 4 6x2+8x−3x−4

STEP 7

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QuestionFactor the following expression completely: $6 x^{2}+5 x-4=$

1. The expression is a quadratic polynomial.
2. We need to factor it into a product of two binomials.

1. Identify the form of the quadratic expression.
2. Use the AC method to factor the quadratic expression.
3. Split the middle term based on the AC method.
4. Factor by grouping.

Identify the form of the quadratic expression. The expression is in the standard form $ax^2 + bx + c$ , where $a = 6$ , $b = 5$ , and $c = -4$ .

Use the AC method to factor the quadratic expression. Calculate the product $ac = 6 \times (-4) = -24$ .

Find two numbers that multiply to $-24$ and add up to $5$ . These numbers are $8$ and $-3$ .
$8 \times (-3) = -24$ $8 + (-3) = 5$

Split the middle term $5x$ using the numbers found:
$6x^2 + 8x - 3x - 4$