Question

exercises
solve each equation by graphing.

1. $x^{2}+7x + 12 = 0$
2. $x^{2}-x - 12 = 0$
3. $x^{2}-4x + 5 = 0$
4. $x^{2}-4x + 5 = 0$
5. $c^{2}+6c + 8 = 0$
6. $a^{2}-2a=-1$
7. $n^{2}-7n=-10$

solve each equation by graphing. if integral roots cannot be found, estimate the roots by stating the consecutive integers between which the roots lie.

1. $p^{2}+4p + 2 = 0$
2. $x^{2}+x - 3 = 0$
3. $d^{2}+6d=-3$
4. $h^{2}+1 = 4h$

Explanation:

Step1: Recall the general form of quadratic function

A quadratic equation $ax^{2}+bx + c=0$ can be graphed as the function $y = ax^{2}+bx + c$. The roots of the equation are the $x$-intercepts of the graph.

Step2: For $x^{2}+7x + 12=0$

Let $y=x^{2}+7x + 12$. We can find the $x$-intercepts by factoring: $x^{2}+7x + 12=(x + 3)(x+4)=0$. The roots are $x=-3$ and $x=-4$. When graphing $y=x^{2}+7x + 12$, the parabola intersects the $x$-axis at $x=-3$ and $x=-4$.

Step3: For $x^{2}-x - 12=0$

Let $y=x^{2}-x - 12$. Factor it: $x^{2}-x - 12=(x - 4)(x+3)=0$. The roots are $x = 4$ and $x=-3$. The graph of $y=x^{2}-x - 12$ intersects the $x$-axis at these points.

Step4: For $x^{2}-4x + 5=0$

Calculate the discriminant $\Delta=b^{2}-4ac$, where $a = 1$, $b=-4$, $c = 5$. $\Delta=(-4)^{2}-4\times1\times5=16 - 20=-4<0$. The graph of $y=x^{2}-4x + 5$ has no $x$-intercepts since the parabola is above the $x$-axis (because $a>0$) and does not cross it.

Step5: For $c^{2}+6c + 8=0$

Let $y=c^{2}+6c + 8$. Factor: $c^{2}+6c + 8=(c + 2)(c+4)=0$. The roots are $c=-2$ and $c=-4$. The graph of $y=c^{2}+6c + 8$ intersects the $c$-axis at these points.

Step6: For $a^{2}-2a=-1$

Rewrite it as $a^{2}-2a + 1=0$. Factor: $(a - 1)^{2}=0$. The root is $a = 1$. The graph of $y=a^{2}-2a + 1$ touches the $a$-axis at $a = 1$.

Step7: For $n^{2}-7n=-10$

Rewrite as $n^{2}-7n + 10=0$. Factor: $n^{2}-7n + 10=(n - 2)(n - 5)=0$. The roots are $n=2$ and $n=5$. The graph of $y=n^{2}-7n + 10$ intersects the $n$-axis at these points.

Step8: For $p^{2}+4p + 2=0$

Calculate the discriminant $\Delta=b^{2}-4ac$ with $a = 1$, $b = 4$, $c = 2$. $\Delta=4^{2}-4\times1\times2=16 - 8 = 8$. Using the quadratic formula $x=\frac{-b\pm\sqrt{\Delta}}{2a}=\frac{-4\pm\sqrt{8}}{2}=-2\pm\sqrt{2}$. $-2-\sqrt{2}\approx-2 - 1.414=-3.414$ and $-2+\sqrt{2}\approx-2 + 1.414=-0.586$. The roots lie between $-4$ and $-3$, and between $-1$ and $0$.

Step9: For $x^{2}+x - 3=0$

Calculate the discriminant $\Delta=b^{2}-4ac$ with $a = 1$, $b = 1$, $c=-3$. $\Delta=1^{2}-4\times1\times(-3)=1 + 12=13$. Using the quadratic formula $x=\frac{-1\pm\sqrt{13}}{2}$. $\frac{-1-\sqrt{13}}{2}\approx\frac{-1-3.606}{2}=-2.303$ and $\frac{-1+\sqrt{13}}{2}\approx\frac{-1 + 3.606}{2}=1.303$. The roots lie between $-3$ and $-2$, and between $1$ and $2$.

Step10: For $d^{2}+6d=-3$

Rewrite as $d^{2}+6d + 3=0$. Calculate the discriminant $\Delta=b^{2}-4ac$ with $a = 1$, $b = 6$, $c = 3$. $\Delta=6^{2}-4\times1\times3=36-12 = 24$. Using the quadratic formula $d=\frac{-6\pm\sqrt{24}}{2}=-3\pm\sqrt{6}$. $-3-\sqrt{6}\approx-3-2.449=-5.449$ and $-3+\sqrt{6}\approx-3 + 2.449=-0.551$. The roots lie between $-6$ and $-5$, and between $-1$ and $0$.

Step11: For $h^{2}+1=4h$

Rewrite as $h^{2}-4h + 1=0$. Calculate the discriminant $\Delta=b^{2}-4ac$ with $a = 1$, $b=-4$, $c = 1$. $\Delta=(-4)^{2}-4\times1\times1=16 - 4=12$. Using the quadratic formula $h=\frac{4\pm\sqrt{12}}{2}=2\pm\sqrt{3}$. $2-\sqrt{3}\approx2 - 1.732 = 0.268$ and $2+\sqrt{3}\approx2+1.732=3.732$. The roots lie between $0$ and $1$, and between $3$ and $4$.

Answer:

1. Roots: $x=-3,x=-4$
2. Roots: $x = 4,x=-3$
3. No real - roots
4. Roots: $c=-2,c=-4$
5. Root: $a = 1$
6. Roots: $n=2,n=5$
7. Roots lie between $-4$ and $-3$, and between $-1$ and $0$
8. Roots lie between $-3$ and $-2$, and between $1$ and $2$
9. Roots lie between $-6$ and $-5$, and between $-1$ and $0$
10. Roots lie between $0$ and $1$, and between $3$ and $4$