Question

1. solve $-3x^2 + 2x + 4 = -x - 3$ by writing a system of equations using the intersection feature of a graphing calculator. round to the nearest hundredth. the graphing calculator shows the curve and line intersect at $x \approx \square$ and $x \approx \square$.

Explanation:

Step1: Form the system of equations

We can set \( y = -3x^2 + 2x + 4 \) (the quadratic function) and \( y=-x - 3\) (the linear function). The solutions to the original equation are the \( x \)-coordinates of the intersection points of these two graphs.

Step2: Solve the quadratic equation

First, rewrite the original equation \( -3x^2+2x + 4=-x - 3\) in standard quadratic form \( ax^{2}+bx + c = 0\).
Add \( x\) and 3 to both sides:
\(-3x^{2}+2x + 4+x + 3=0\)
Combine like terms:
\(-3x^{2}+(2x + x)+(4 + 3)=0\)
\(-3x^{2}+3x + 7 = 0\)
Multiply both sides by - 1 to make the coefficient of \(x^{2}\) positive:
\(3x^{2}-3x - 7=0\)

Now we use the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\), where \(a = 3\), \(b=-3\) and \(c=-7\).

First, calculate the discriminant \(\Delta=b^{2}-4ac\):
\(\Delta=(-3)^{2}-4\times3\times(-7)=9 + 84=93\)

Then, find the roots:
\(x=\frac{-(-3)\pm\sqrt{93}}{2\times3}=\frac{3\pm\sqrt{93}}{6}\)

Calculate the numerical values:
\(\sqrt{93}\approx9.6437\)
For the plus sign:
\(x_1=\frac{3 + 9.6437}{6}=\frac{12.6437}{6}\approx2.11\)
For the minus sign:
\(x_2=\frac{3-9.6437}{6}=\frac{-6.6437}{6}\approx - 1.11\)

Answer:

\(x\approx - 1.11\) and \(x\approx2.11\)