Question Solve for $y$ using factoring or the quadratic formula: $y^{2} - 6y + 7 = 0$

Studdy Solution

STEP 1

Assumptions
1. We have a quadratic equation in the form $y^2 - 6y + 7 = 0$ .
2. We will first attempt to solve the equation by factoring.
3. If factoring is not possible or is difficult, we will use the quadratic formula.

STEP 2

Attempt to factor the quadratic equation. We look for two numbers that multiply to give the constant term (7) and add to give the coefficient of the linear term (-6).

STEP 3

Check if there are integer factors of 7 that add up to -6. The factors of 7 are 1 and 7, and neither -1 and -7 nor 1 and 7 add up to -6.

STEP 4

Since we cannot find factors that satisfy the conditions for factoring, we conclude that factoring is not possible or is too difficult for this quadratic equation.

STEP 5

We will now use the quadratic formula to solve for $y$ . The quadratic formula is given by:
$y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
where $a$ , $b$ , and $c$ are the coefficients from the quadratic equation $ay^2 + by + c = 0$ .

STEP 6

Identify the coefficients $a$ , $b$ , and $c$ from the given quadratic equation $y^2 - 6y + 7 = 0$ .
$a = 1, \quad b = -6, \quad c = 7$

STEP 7

Substitute the values of $a$ , $b$ , and $c$ into the quadratic formula.
$y = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 7}}{2 \cdot 1}$

STEP 8

Simplify the expression inside the square root.
$y = \frac{6 \pm \sqrt{36 - 28}}{2}$

STEP 9

Continue simplifying the expression under the square root.
$y = \frac{6 \pm \sqrt{8}}{2}$

STEP 10

Simplify the square root of 8 by expressing it as $2\sqrt{2}$ .
$y = \frac{6 \pm 2\sqrt{2}}{2}$

STEP 11

Divide both terms in the numerator by 2.
$y = \frac{6}{2} \pm \frac{2\sqrt{2}}{2}$

STEP 12

Simplify the fractions.
$y = 3 \pm \sqrt{2}$

STEP 13

Write the final solutions for $y$ .
$y = 3 + \sqrt{2} \quad \text{or} \quad y = 3 - \sqrt{2}$
The solutions for the equation $y^2 - 6y + 7 = 0$ are $y = 3 + \sqrt{2}$ and $y = 3 - \sqrt{2}$ .

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