Solve the Quadratic Equation: -x² + 10x - 21 = 0

To solve the equation $-x^2 + 10x - 21 = 0$, we will use the quadratic formula. The equation is in the form $ax^2 + bx + c = 0$ where:

• $a = -1$
• $b = 10$
• $c = -21$

First, compute the discriminant, which is given by $b^2 - 4ac$:

$b^2 - 4ac = (10)^2 - 4(-1)(-21) = 100 - 84 = 16$

Since the discriminant is positive, we have two distinct real solutions. We apply the quadratic formula:

$x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} = \frac{{-10 \pm \sqrt{16}}}{-2}$

Calculate the roots:

• First root: $x_1 = \frac{{-10 + 4}}{-2} = \frac{{-6}}{-2} = 3$
• Second root: $x_2 = \frac{{-10 - 4}}{-2} = \frac{{-14}}{-2} = 7$

Therefore, the solutions to the equation are $x_1 = 7$ and $x_2 = 3$.

The correct choice from the options provided is:

$x_1=7,x_2=3$

Thus, the solutions to the quadratic equation are $\mathbf{x_1 = 7}$ and $\mathbf{x_2 = 3}$.