Answered

Solve $0=4 x^2+12 x+9$.



Select the equation that shows the correct substitution of $a, b$, and $c$ in the quadratic formula.

A$x=\frac{12 \pm \sqrt{12^2-4(4)(9)}}{2(4)}$
B$x=\frac{-12 \pm \sqrt{12^2-4(4)(9)}}{2(4)}$
C$x=\frac{-12 \pm \sqrt{12^2+4(4)(9)}}{2(4)}$



Simplify the expression to solve the equation.

$x=$

Verified Answer :

  • Identify coefficients: \(a=4\), \(b=12\), and \(c=9\).
  • Select the correct substitution: \(x = \frac{-12 \pm \sqrt{12^2 - 4(4)(9)}}{2(4)}\).
  • Calculate the discriminant: \(12^2 - 4(4)(9) = 0\).
  • Determine the root: \(x = \boxed{-1.5}\).

Explanation

  1. Identifying Coefficients and Substitution

The quadratic equation is given as \(4x^2 + 12x + 9 = 0\). To solve this, we identify the coefficients for the standard quadratic form \(ax^2 + bx + c = 0\). Here, \(a = 4\), \(b = 12\), and \(c = 9\). The quadratic formula states that the roots are given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substituting our values, we get:
\[x = \frac{-12 \pm \sqrt{12^2 - 4(4)(9)}}{2(4)}\]
This corresponds to the second given option.

  1. Solving the Equation

Now, we simplify the expression by calculating the discriminant \(D = b^2 - 4ac\):
\[D = 12^2 - 4 \times 4 \times 9 = 144 - 144 = 0\]
Since the discriminant is 0, the equation has one repeated root. Substituting this back into the formula:
\[x = \frac{-12 \pm \sqrt{0}}{2(4)} = \frac{-12}{8}\]
Dividing numerator and denominator by 4:
\[x = -\frac{3}{2} = -1.5\]
Thus, the equation has a single solution \(x = -1.5\).

Examples

Quadratic equations are fundamental in modeling real-world situations, such as determining the time at which a projectile launched from the ground reaches a specific height or calculating the dimensions of an area to minimize or maximize resources. For example, if the height of a ball tossed upwards is modeled by a quadratic function, finding when it hits the ground involves solving for the roots of that function using the same quadratic formula approach demonstrated here. Mastering this allows one to predict future states in various physical systems.

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