Question Solve the equation $z^{2} + 9 = 10z$ .

Studdy Solution

STEP 1

Assumptions
1. We are solving a quadratic equation in the form of $z^2 + bz + c = 0$ .
2. The given equation is $z^2 + 9 = 10z$ .
3. We will rearrange the equation into standard quadratic form.
4. We will use the quadratic formula to find the solutions for $z$ if necessary.

STEP 2

Rearrange the given equation into standard quadratic form by moving all terms to one side of the equation.
$z^2 - 10z + 9 = 0$

STEP 3

Identify the coefficients $a$ , $b$ , and $c$ from the standard form $az^2 + bz + c = 0$ .
Here, $a = 1$ , $b = -10$ , and $c = 9$ .

STEP 4

Check if the quadratic can be factored easily. If so, factor the quadratic equation to find the solutions for $z$ .

STEP 5

Attempt to factor the quadratic equation by finding two numbers that multiply to $ac$ (which is $1 \times 9 = 9$ ) and add up to $b$ (which is $-10$ ).

STEP 6

The two numbers that satisfy the conditions from STEP_5 are $-1$ and $-9$ because $-1 \times -9 = 9$ and $-1 + (-9) = -10$ .

STEP 7

Write the factored form of the quadratic equation using the two numbers found in STEP_6.
$(z - 1)(z - 9) = 0$

STEP 8

Apply the zero product property, which states that if the product of two factors is zero, then at least one of the factors must be zero.

STEP 9

Set each factor equal to zero and solve for $z$ .
$z - 1 = 0 \quad \text{and} \quad z - 9 = 0$

STEP 10

Solve the first equation for $z$ .
$z = 1$

STEP 11

Solve the second equation for $z$ .
$z = 9$
The solutions to the equation $z^2 + 9 = 10z$ are $z = 1$ and $z = 9$ .

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