find the range of $k$ for the equation $x^2 - 2kx + k^2 - 2k = 6$ has real roots. Find the roots in terms of $k$

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Algebra 1

Domain and range of quadratic functions: equations

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Q. find the range of

k

for the equation

x^2 - 2kx + k^2 - 2k = 6

has real roots. Find the roots in terms of

k

Calculate Discriminant: To find the range of $k$ for which the quadratic equation has real roots, we need to ensure that the discriminant of the equation is greater than or equal to zero. The discriminant ( $D$ ) of a quadratic equation $ax^2 + bx + c = 0$ is given by $D = b^2 - 4ac$ .
Simplify Equation: First, let's rewrite the equation in the standard quadratic form: $x^2 - 2kx + (k^2 - 2k - 6) = 0$ . Here, $a = 1$ , $b = -2k$ , and $c = k^2 - 2k - 6$ .
Check Discriminant: Now, calculate the discriminant: $D = (-2k)^2 - 4(1)(k^2 - 2k - 6)$ .
Solve Inequality: Simplify the discriminant: $D = 4k^2 - 4(k^2 - 2k - 6) = 4k^2 - 4k^2 + 8k + 24$ .
Find Roots: Further simplification gives us: $D = 8k + 24$ .
Find Roots: Further simplification gives us: $D = 8k + 24$ .For the equation to have real roots, the discriminant must be greater than or equal to zero: $8k + 24 \geq 0$ .
Find Roots: Further simplification gives us: $D = 8k + 24$ .For the equation to have real roots, the discriminant must be greater than or equal to zero: $8k + 24 \geq 0$ .Solve the inequality for $k$ : $8k \geq -24$ .
Find Roots: Further simplification gives us: $D = 8k + 24$ .For the equation to have real roots, the discriminant must be greater than or equal to zero: $8k + 24 \geq 0$ .Solve the inequality for $k$ : $8k \geq -24$ .Divide both sides by $8$ to isolate $k$ : $k \geq -3$ .
Find Roots: Further simplification gives us: $D = 8k + 24$ .For the equation to have real roots, the discriminant must be greater than or equal to zero: $8k + 24 \geq 0$ .Solve the inequality for $k$ : $8k \geq -24$ .Divide both sides by $8$ to isolate $k$ : $k \geq -3$ .Now that we have the range of $k$ , we can find the roots of the equation in terms of $k$ using the quadratic formula, $x = [-b \pm \sqrt{D}] / (2a)$ , where $8k + 24 \geq 0$ $0$ is the discriminant.
Find Roots: Further simplification gives us: $D = 8k + 24$ . For the equation to have real roots, the discriminant must be greater than or equal to zero: $8k + 24 \geq 0$ . Solve the inequality for $k$ : $8k \geq -24$ . Divide both sides by $8$ to isolate $k$ : $k \geq -3$ . Now that we have the range of $k$ , we can find the roots of the equation in terms of $k$ using the quadratic formula, $x = \frac{-b \pm \sqrt{D}}{2a}$ , where $8k + 24 \geq 0$ $0$ is the discriminant. Substitute $8k + 24 \geq 0$ $1$ , $8k + 24 \geq 0$ $2$ , and $D = 8k + 24$ into the quadratic formula: $8k + 24 \geq 0$ $4$ .
Find Roots: Further simplification gives us: $D = 8k + 24$ .For the equation to have real roots, the discriminant must be greater than or equal to zero: $8k + 24 \geq 0$ .Solve the inequality for $k$ : $8k \geq -24$ .Divide both sides by $8$ to isolate $k$ : $k \geq -3$ .Now that we have the range of $k$ , we can find the roots of the equation in terms of $k$ using the quadratic formula, $x = [-b \pm \sqrt{D}] / (2a)$ , where $8k + 24 \geq 0$ $0$ is the discriminant.Substitute $8k + 24 \geq 0$ $1$ , $8k + 24 \geq 0$ $2$ , and $D = 8k + 24$ into the quadratic formula: $8k + 24 \geq 0$ $4$ .Simplify the expression for the roots: $8k + 24 \geq 0$ $5$ .
Find Roots: Further simplification gives us: $D = 8k + 24$ . For the equation to have real roots, the discriminant must be greater than or equal to zero: $8k + 24 \geq 0$ . Solve the inequality for $k$ : $8k \geq -24$ . Divide both sides by $8$ to isolate $k$ : $k \geq -3$ . Now that we have the range of $k$ , we can find the roots of the equation in terms of $k$ using the quadratic formula, $x = [-b \pm \sqrt{D}] / (2a)$ , where $8k + 24 \geq 0$ $0$ is the discriminant. Substitute $8k + 24 \geq 0$ $1$ , $8k + 24 \geq 0$ $2$ , and $D = 8k + 24$ into the quadratic formula: $8k + 24 \geq 0$ $4$ . Simplify the expression for the roots: $8k + 24 \geq 0$ $5$ . The roots in terms of $k$ are $8k + 24 \geq 0$ $7$ .