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Math  /  Algebra

QuestionFactor or take roots 7) 2x2+2=662 x^{2}+2=66 8) 6n2−3=1896 n^{2}-3=189

Studdy Solution

STEP 1

1. The problems involve quadratic equations that need to be solved for their roots.
2. The equations can be simplified and solved using algebraic techniques such as factoring or using the quadratic formula.
3. The equations may need to be rearranged to standard quadratic form ax2+bx+c=0 ax^2 + bx + c = 0 .

STEP 2

1. Simplify and rearrange the equation to standard quadratic form.
2. Solve the quadratic equation by factoring or using the quadratic formula.
3. Verify the solutions.
**Problem 7: 2x2+2=66 2x^2 + 2 = 66 **

STEP 3

Simplify and rearrange the equation to standard quadratic form. Subtract 66 from both sides:
2x2+2=66 2x^2 + 2 = 66 2x2+2−66=0 2x^2 + 2 - 66 = 0 2x2−64=0 2x^2 - 64 = 0

STEP 4

Factor the quadratic equation. First, factor out the greatest common factor from the terms:
2(x2−32)=0 2(x^2 - 32) = 0

STEP 5

Solve for x x . Since 2≠0 2 \neq 0 , we set the remaining factor equal to zero:
x2−32=0 x^2 - 32 = 0

STEP 6

Solve for x x by taking the square root of both sides:
x2=32 x^2 = 32 x=±32 x = \pm \sqrt{32} x=±42 x = \pm 4\sqrt{2}

STEP 7

Verify the solutions by substituting back into the original equation:
For x=42 x = 4\sqrt{2} and x=−42 x = -4\sqrt{2} , check:
2(42)2+2=66 2(4\sqrt{2})^2 + 2 = 66 2(−42)2+2=66 2(-4\sqrt{2})^2 + 2 = 66
Both simplify to 66 66 , confirming the solutions.
**Problem 8: 6n2−3=189 6n^2 - 3 = 189 **
STEP_1: Simplify and rearrange the equation to standard quadratic form. Subtract 189 from both sides:
6n2−3=189 6n^2 - 3 = 189 6n2−3−189=0 6n^2 - 3 - 189 = 0 6n2−192=0 6n^2 - 192 = 0
STEP_2: Factor the quadratic equation. First, factor out the greatest common factor from the terms:
6(n2−32)=0 6(n^2 - 32) = 0
STEP_3: Solve for n n . Since 6≠0 6 \neq 0 , we set the remaining factor equal to zero:
n2−32=0 n^2 - 32 = 0
STEP_4: Solve for n n by taking the square root of both sides:
n2=32 n^2 = 32 n=±32 n = \pm \sqrt{32} n=±42 n = \pm 4\sqrt{2}
STEP_5: Verify the solutions by substituting back into the original equation:
For n=42 n = 4\sqrt{2} and n=−42 n = -4\sqrt{2} , check:
6(42)2−3=189 6(4\sqrt{2})^2 - 3 = 189 6(−42)2−3=189 6(-4\sqrt{2})^2 - 3 = 189
Both simplify to 189 189 , confirming the solutions.
The solutions for both problems are: Problem 7: x=±42 x = \pm 4\sqrt{2} Problem 8: n=±42 n = \pm 4\sqrt{2}

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