$$ 9 x^{2}+24 x+16=0 $$ 2 of $$ \mathbf{2} $$ : Use the discriminant. $$ b^{2}-4 a c $$, to determine the number of solutions of the given quadratic equation. Then solve the quadratic equation using th iula $$ x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} $$. Keyboard Si at the number and type of solutions. Then, enter the solutions. Previous Step cting an option will display any text boxes needed to complete your answer. Two different real solutions One repeated real solution

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UpStudy AI · Accepted Answer

**Step 1: Identify the coefficients.**  
The quadratic equation is  
$$
9x^2 + 24x + 16 = 0.
$$  
Here,  
$$
a = 9, \quad b = 24, \quad c = 16.
$$

**Step 2: Compute the discriminant.**  
The discriminant is given by  
$$
\Delta = b^2 - 4ac.
$$  
Substitute the values:  
$$
\Delta = 24^2 - 4 \cdot 9 \cdot 16 = 576 - 576 = 0.
$$

**Step 3: Determine the number of solutions.**  
Since  
$$
\Delta = 0,
$$  
there is one repeated (real) solution.

**Step 4: Solve the quadratic equation using the quadratic formula.**  
The quadratic formula is  
$$
x = \frac{-b \pm \sqrt{\Delta}}{2a}.
$$  
Substitute the known values:  
$$
x = \frac{-24 \pm \sqrt{0}}{2 \cdot 9} = \frac{-24 \pm 0}{18}.
$$

**Step 5: Simplify to find the solution.**  
$$
x = \frac{-24}{18} = -\frac{4}{3}.
$$

**Final Answer:**  
The given quadratic equation has one repeated real solution,  
$$
x = -\frac{4}{3}.
$$
The quadratic equation $$9x^{2} + 24x + 16 = 0$$ has one repeated real solution: $$x = -\frac{4}{3}$$.

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