+0

# Create an equation that has the three solutions x=2,4, and 5.

0
370
1

Create an equation that has the three solutions x=2,4, and 5.

Aug 12, 2017

#1
+1884
+1

To find an equation that has the three solutions of 2, 4, and 5, first set each solution equal to x.

$$x=2$$

$$x=4$$

$$x=5$$

Next, figure out what number goes with each of the three solutions that makes x equal to zero on the right side.  For x=2, the number will be -2; for x=4, the number is -4; for x=5, the number will be -5. What you do on the right side, you do to the left side as well.

$$x=2$$

$$x-2=2-2$$

$$x-2=0$$

$$x=4$$

$$x-4=4-4$$

$$x-4=0$$

$$x=5$$

$$x-5=5-5$$

$$x-5=0$$

Next, combine all three equations using the zero product.

$$(x-2)\times(x-4)\times(x-5)=0$$

Next, expand the left side.

$$(x-2)\times(x-4)\times(x-5)=0$$

$$({x}^{2}-4x-2x+8)\times(x-5)=0$$

$$({x}^{2}-6x+8)\times(x-5)=0$$

$${x}^{3}-5{x}^{2}-6{x}^{2}+30x+8x-40=0$$

$${x}^{3}-11{x}^{2}+38x-40=0$$

Next, replace 0 on the right side with y and then switch the equation around so that the y is on the left side.

$${x}^{3}-11{x}^{2}+38x-40=y$$

$$y={x}^{3}-11{x}^{2}+38x-40$$

.
Aug 12, 2017
edited by gibsonj338  Aug 12, 2017

#1
+1884
+1

To find an equation that has the three solutions of 2, 4, and 5, first set each solution equal to x.

$$x=2$$

$$x=4$$

$$x=5$$

Next, figure out what number goes with each of the three solutions that makes x equal to zero on the right side.  For x=2, the number will be -2; for x=4, the number is -4; for x=5, the number will be -5. What you do on the right side, you do to the left side as well.

$$x=2$$

$$x-2=2-2$$

$$x-2=0$$

$$x=4$$

$$x-4=4-4$$

$$x-4=0$$

$$x=5$$

$$x-5=5-5$$

$$x-5=0$$

Next, combine all three equations using the zero product.

$$(x-2)\times(x-4)\times(x-5)=0$$

Next, expand the left side.

$$(x-2)\times(x-4)\times(x-5)=0$$

$$({x}^{2}-4x-2x+8)\times(x-5)=0$$

$$({x}^{2}-6x+8)\times(x-5)=0$$

$${x}^{3}-5{x}^{2}-6{x}^{2}+30x+8x-40=0$$

$${x}^{3}-11{x}^{2}+38x-40=0$$

Next, replace 0 on the right side with y and then switch the equation around so that the y is on the left side.

$${x}^{3}-11{x}^{2}+38x-40=y$$

$$y={x}^{3}-11{x}^{2}+38x-40$$

gibsonj338 Aug 12, 2017
edited by gibsonj338  Aug 12, 2017