Practice quadratic equations and check every root

1. 1
   Factor

   Use the pair negative 4 and negative 5.

   $$(x-4)(x-5)=0$$
2. 2
   Use the zero-product property

   At least one factor must be zero.

   $$x=4\quad\text{or}\quad x=5$$

Independent check: Both values make the original polynomial zero, and their sum and product are 9 and 20.

1. 1
   Factor the quadratic

   The cross terms four x and negative three x combine to x.

   $$(2x-3)(x+2)=0$$
2. 2
   Solve each factor

   Set both factors equal to zero.

   $$x=\frac32\quad\text{or}\quad x=-2$$

Independent check: Substitution gives zero for each root, and their product is c/a = -3.

1. 1
   Recognize the square

   Nine is 3 squared and the middle term is twice 3x.

   $$x^2+6x+9=(x+3)^2$$
2. 2
   Solve and count

   The same factor occurs twice.

   $$(x+3)^2=0\Rightarrow x=-3\text{ with multiplicity }2$$

Independent check: The discriminant is zero and substitution of negative 3 gives zero.

1. 1
   Compute the discriminant

   Use b squared minus 4ac.

   $$\Delta=4^2-4(1)(8)=-16$$
2. 2
   Apply the formula

   Use the principal square root of negative 16 as 4i.

   $$x=\frac{-4\pm4i}{2}=-2\pm2i$$

Independent check: The roots have sum negative 4 and product 8, matching the coefficients.

1. 1
   Isolate the square

   Add 12 and divide by 3.

   $$x^2=4$$
2. 2
   Take both square roots

   Positive 2 and negative 2 both square to 4.

   $$x=\pm2$$

Independent check: Each root makes 3x squared minus 12 equal zero.

1. 1
   Compute the discriminant

   Subtract four times a times c from b squared.

   $$\Delta=(-5)^2-4(1)(-1)=29$$
2. 2
   Apply the formula

   Use negative b over twice a.

   $$x=\frac{5\pm\sqrt{29}}{2}$$

Independent check: The two roots have sum 5 and product negative 1.

1. 1
   Simplify and factor

   Divide by negative 16, then factor the monic quadratic.

   $$t^2-4t-5=(t-5)(t+1)=0$$
2. 2
   Interpret the roots

   Reject negative elapsed time in this model.

   $$t=5\quad\text{or}\quad t=-1$$

Independent check: Substitution gives negative 400 plus 320 plus 80, which equals zero.

1. 1
   Complete the square

   Add and subtract 16 inside the expression.

   $$x^2-8x+7=(x-4)^2-9$$
2. 2
   Read the vertex

   The squared term is smallest when x equals 4.

   $$f(4)=-9$$

Independent check: The axis formula gives x = 4, and nearby values produce larger outputs.

1. 1
   Convert roots to factors

   Subtract each root from x.

   $$(x+3)(x-7)=0$$
2. 2
   Expand

   Multiply the binomials and combine the middle terms.

   $$x^2-4x-21=0$$

Independent check: Substitution of negative 3 and 7 makes the polynomial zero.

1. 1
   Set the discriminant to zero

   A repeated real root occurs only when the discriminant is zero.

   $$4^2-4(1)k=0$$
2. 2
   Solve for k

   Divide 16 = 4k by 4.

   $$k=4$$

Independent check: With k = 4, the equation becomes (x + 2) squared equals zero.