The equation $$ 12 t^{2}+13 t-4=0 $$ has solutions of the form $$ t=\frac{N \pm \sqrt{D}}{M} $$ (A) Use the quadratic formula to solve this equation and find the appropriate integer values of $$ N, M $$, and $$ D $$. Do not worry about simplifying the $$ \sqrt{D} $$ yet in this part of the problem. N $$ N=\square $$; $$ D=\square $$

Question

UpStudy AI · Accepted Answer

Solve the equation $$ 12t^{2}+13t-4=0 $$.
Solve the quadratic equation by following steps:
- step0: Solve by factoring:
  $$12t^{2}+13t-4=0$$
- step1: Factor the expression:
  $$\left(3t+4\right)\left(4t-1\right)=0$$
- step2: Separate into possible cases:
  $$\begin{align}&4t-1=0\&3t+4=0\end{align}$$
- step3: Solve the equation:
  $$\begin{align}&t=\frac{1}{4}\&t=-\frac{4}{3}\end{align}$$
- step4: Rewrite:
  $$t_{1}=-\frac{4}{3},t_{2}=\frac{1}{4}$$
The solutions to the equation $$12t^{2}+13t-4=0$$ are $$t_{1}=-\frac{4}{3}$$ and $$t_{2}=\frac{1}{4}$$.

To find the appropriate integer values of $$N$$, $$M$$, and $$D$$ using the quadratic formula, we can compare the solutions obtained with the general form of the quadratic formula:

$$ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

In this case, $$a = 12$$, $$b = 13$$, and $$c = -4$$.

Comparing the solutions with the general form, we can see that:

- $$N = -b = -13$$
- $$D = b^2 - 4ac = 13^2 - 4(12)(-4) = 169 + 192 = 361$$
- $$M = 2a = 2(12) = 24$$

Therefore, the appropriate integer values are $$N = -13$$, $$D = 361$$, and $$M = 24$$.
$$ N = -13 $$; $$ D = 361 $$

The equation has solutions of the form

(A) Use the quadratic formula to solve this equation and find the
appropriate integer values of , and . Do not worry about
simplifying the yet in this part of the problem.
N ;

Upstudy AI Solution

Answer

Solution

Mind Expander

Related Questions

Latest Algebra Questions

The equation has solutions of the form (A) Use the quadratic formula to solve this equation and find the appropriate integer values of , and . Do not worry about simplifying the yet in this part of the problem. N ;