Math

Question Solve the quadratic equation z213z+12=0z^{2} - 13z + 12 = 0 for real values of zz.

Studdy Solution

STEP 1

1. The expression z213z+12z^{2}-13z+12 is a quadratic polynomial in the variable zz.
2. The quadratic can be factored or solved using the quadratic formula to find its roots.
3. The roots are the values of zz that satisfy the equation z213z+12=0z^{2}-13z+12=0.

STEP 2

1. Factor the quadratic polynomial.
2. Verify the roots by plugging them back into the original equation.

STEP 3

Attempt to factor the quadratic polynomial z213z+12z^{2}-13z+12 by finding two numbers that multiply to 1212 and add up to 13-13.

STEP 4

Identify the numbers as 1-1 and 12-12, since (1)×(12)=12(-1) \times (-12) = 12 and (1)+(12)=13(-1) + (-12) = -13.

STEP 5

Write the quadratic polynomial as a product of two binomials using the identified numbers:
z213z+12=(z1)(z12) z^{2}-13z+12 = (z-1)(z-12)

STEP 6

Verify the roots by plugging z=1z=1 into the original equation:
(1)213(1)+12=113+12=0 (1)^2 - 13(1) + 12 = 1 - 13 + 12 = 0

STEP 7

Verify the second root by plugging z=12z=12 into the original equation:
(12)213(12)+12=144156+12=0 (12)^2 - 13(12) + 12 = 144 - 156 + 12 = 0
The roots of the quadratic polynomial z213z+12z^{2}-13z+12 are z=1z=1 and z=12z=12.

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