Math  /  Algebra

QuestionFind all real solutions of the equation 3(t5)27=173(t-5)^{2}-7=17
Write out the exact answers (no decimal values), with answers separated by a comma. t=t=

Studdy Solution

STEP 1

1. The equation 3(t5)27=173(t-5)^2 - 7 = 17 is a quadratic equation in terms of (t5)(t-5).
2. Solving the equation involves isolating the quadratic term and then using algebraic methods to find the values of tt.

STEP 2

1. Isolate the quadratic term.
2. Solve for the squared term.
3. Solve for tt by considering both the positive and negative square roots.
4. Write the solutions in exact form.

STEP 3

First, isolate the quadratic term 3(t5)23(t-5)^2 by adding 77 to both sides of the equation:
3(t5)27+7=17+7 3(t-5)^2 - 7 + 7 = 17 + 7 3(t5)2=24 3(t-5)^2 = 24

STEP 4

Next, divide both sides by 33 to solve for (t5)2(t-5)^2:
3(t5)23=243 \frac{3(t-5)^2}{3} = \frac{24}{3} (t5)2=8 (t-5)^2 = 8

STEP 5

Take the square root of both sides to solve for t5t-5. Remember to consider both the positive and negative square roots:
t5=8ort5=8 t-5 = \sqrt{8} \quad \text{or} \quad t-5 = -\sqrt{8}

STEP 6

Solve for tt by adding 55 to both sides of each equation:
t=5+8ort=58 t = 5 + \sqrt{8} \quad \text{or} \quad t = 5 - \sqrt{8}

STEP 7

Express the solutions in exact form. Simplify 8\sqrt{8} as 4×2=22\sqrt{4 \times 2} = 2\sqrt{2}:
t=5+22,t=522 t = 5 + 2\sqrt{2}, \quad t = 5 - 2\sqrt{2}
The real solutions are:
t=5+22,t=522 t = 5 + 2\sqrt{2}, \quad t = 5 - 2\sqrt{2}

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