방정식과 부등식 전체 문제

For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.

${x}^{2}+4x+7=0$

For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.

$3{x}^{2}+5x-8=0$

For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.

$9{x}^{2}-30x+25=0$

For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.

$2{x}^{2}-3x-7=0$

For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.

$6{x}^{2}-x-2=0$

For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution.

$2{x}^{2}+5x+3=0$

For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution.

${x}^{2}+x=4$

For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution.

$2{x}^{2}-8x-5=0$

For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution.

$3{x}^{2}-5x+1=0$

For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution.

${x}^{2}+4x+2=0$

For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution.

$4+\frac{1}{x}-\frac{1}{{x}^{2}}=0$

For the following exercises, enter the expressions into your graphing utility and find the zeroes to the equation (the x-intercepts) by using 2^nd CALC 2:zero. Recall finding zeroes will ask left bound (move your cursor to the left of the zero,enter), then right bound (move your cursor to the right of the zero,enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth.

${\text{Y}}_{1}=4{x}^{2}+3x-2$

For the following exercises, enter the expressions into your graphing utility and find the zeroes to the equation (the x-intercepts) by using 2^nd CALC 2:zero. Recall finding zeroes will ask left bound (move your cursor to the left of the zero,enter), then right bound (move your cursor to the right of the zero,enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth.

${\text{Y}}_{1}=-3{x}^{2}+8x-1$

For the following exercises, enter the expressions into your graphing utility and find the zeroes to the equation (the x-intercepts) by using 2^nd CALC 2:zero. Recall finding zeroes will ask left bound (move your cursor to the left of the zero,enter), then right bound (move your cursor to the right of the zero,enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth.

${\text{Y}}_{1}=0.5{x}^{2}+x-7$

For the following exercises, enter the expressions into your graphing utility and find the zeroes to the equation (the x-intercepts) by using 2^nd CALC 2:zero. Recall finding zeroes will ask left bound (move your cursor to the left of the zero,enter), then right bound (move your cursor to the right of the zero,enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth.

To solve the quadratic equation ${x}^{2}+5x-7=4,$ we can graph these two equations

$\begin{array}{c} \\ \begin{array}{c}{\text{Y}}_{1}={x}^{2}+5x-7 \\ {\text{Y}}_{2}=4\end{array}\end{array}$

and find the points of intersection. Recall 2^nd CALC 5:intersection. Do this and find the solutions to the nearest tenth.

For the following exercises, enter the expressions into your graphing utility and find the zeroes to the equation (the x-intercepts) by using 2^nd CALC 2:zero. Recall finding zeroes will ask left bound (move your cursor to the left of the zero,enter), then right bound (move your cursor to the right of the zero,enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth.

To solve the quadratic equation $0.3{x}^{2}+2x-4=2,$ we can graph these two equations

$\begin{array}{c} \\ \begin{array}{c}{\text{Y}}_{1}=0.3{x}^{2}+2x-4 \\ {\text{Y}}_{2}=2\end{array}\end{array}$

and find the points of intersection. Recall 2^nd CALC 5:intersection. Do this and find the solutions to the nearest tenth.

Beginning with the general form of a quadratic equation, $a{x}^{2}+bx+c=0,$ solve for x by using the completing the square method, thus deriving the quadratic formula.

Show that the sum of the two solutions to the quadratic equation is $-\frac{b}{a}$ .

A person has a garden that has a length 10 feet longer than the width. Set up a quadratic equation to find the dimensions of the garden if its area is 119 ft.². Solve the quadratic equation to find the length and width.

Abercrombie and Fitch stock had a price given as $P=0.2{t}^{2}-5.6t+50.2,$ where $t$ is the time in months from 1999 to 2001. ( $t=1$ is January 1999). Find the two months in which the price of the stock was $30.