The Quadratic Formula: Complete the equation

To solve this problem, we need to find the value of $C$ in the quadratic equation $4x^2 + 5x + C = 0$ such that the equation has no real solution. This occurs when the discriminant of the quadratic equation is less than zero.

The discriminant $\Delta$ of a quadratic equation $ax^2 + bx + c = 0$ is given by:

$\Delta = b^2 - 4ac$

In our equation:

• $a = 4$
• $b = 5$
• $c = C$

The discriminant becomes:

$\Delta = 5^2 - 4 \cdot 4 \cdot C = 25 - 16C$

For the quadratic equation to have no real solutions, the discriminant must be less than zero:

$25 - 16C < 0$

Solving this inequality for $C$:

$25 < 16C$

$\frac{25}{16} < C$

Therefore, the condition for $C$ is that it must be greater than $\frac{25}{16}$ for the quadratic equation to have no real solutions.

Therefore, the correct answer is $\frac{25}{16} < C$.

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: $x = \frac{4}{3}$, $\square = -\frac{9}{8}$.

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

• Step 1: Substitute the given solution $x = -1$ into the equation.
• Step 2: Simplify the equation and solve for the missing coefficient.

Now, let's work through each step:

Step 1: Substitute $x = -1$ into the equation $7x^2 + \square x + 2 = 0$.
This gives us:
$7(-1)^2 + \square(-1) + 2 = 0$.

Step 2: Simplify the equation.
We know that $(-1)^2 = 1$, so:
$7 \times 1 - \square + 2 = 0$.

This simplifies to:
$7 - \square + 2 = 0$,
which simplifies further to:
$9 - \square = 0$.

Solving for the blank, we have:
$\square = 9$.

Therefore, the missing coefficient is $9$.