Solve: Finding the Missing Coefficient in x²+3x-2=0 for One Solution

Complete the equation so that it has only one solution and then solve.

$\square x^2+3x-2=0$

To solve this problem, we'll follow these steps:

• Step 1: Set up the discriminant equation for the quadratic to have one solution.
• Step 2: Solve for $a$ using the condition $\Delta = 0$.
• Step 3: Once $a$ is determined, solve for $x$ using the quadratic formula.

Step 1: To have only one solution, we use the discriminant condition:

$\Delta = b^2 - 4ac = 0$

Substitute $b = 3$ and $c = -2$:

$3^2 - 4a(-2) = 0$

Simplify the equation:

$9 + 8a = 0$

Step 2: Solve for $a$:

$8a = -9$

$a = -\frac{9}{8}$

Step 3: Substitute $a = -\frac{9}{8}$ in the quadratic equation and use the quadratic formula:

Our equation becomes $-\frac{9}{8}x^2 + 3x - 2 = 0$.

The quadratic formula is:

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

Using $a = -\frac{9}{8}$, $b = 3$, and $c = -2$:

$x = \frac{-3 \pm \sqrt{3^2 - 4(-\frac{9}{8})(-2)}}{2(-\frac{9}{8})}$

Since the discriminant is 0, we have one solution:

$x = \frac{-3}{-3}$

Thus, the solution is:

$x = \frac{4}{3}$, and the required value of $\square = a = -\frac{9}{8}$.

Therefore, the correct choice is: $x = \frac{4}{3}$, $\square = -\frac{9}{8}$.