Solve the quadratic equation using completing the square. $$ t^{2}=10 t+19 $$

Question

UpStudy AI · Accepted Answer

We are given the quadratic equation

t² = 10t + 19.

Step 1. First, bring all terms to one side by subtracting 10t + 19 from both sides:

t² - 10t - 19 = 0.

Step 2. Isolate the constant term:

t² - 10t = 19.

Step 3. To complete the square, take the coefficient of t, which is -10, divide it by 2, and then square it.

Half of -10 is -5, and (-5)² = 25.

Add 25 to both sides of the equation:

t² - 10t + 25 = 19 + 25.

Step 4. Now the left-hand side is a perfect square:

(t - 5)² = 44.

Step 5. Taking the square root of both sides yields:

t - 5 = ±√44.

Step 6. Simplify √44. Since 44 = 4 × 11, we have:

√44 = √(4×11) = 2√11.

So, the equation becomes:

t - 5 = ±2√11.

Step 7. Finally, solve for t by adding 5 to both sides:

t = 5 ± 2√11.

Thus, the solutions to the quadratic equation using completing the square are:

t = 5 + 2√11  and  t = 5 - 2√11.
The solutions are $$ t = 5 + 2\sqrt{11} $$ and $$ t = 5 - 2\sqrt{11} $$.

Solve the quadratic equation using completing the square. \( t^{2}=10 t+19 \)