Question

There are 5 quadratics below. Four of them have two distinct roots each. The other has only one distinct root; find the value of that root.a. 4x^2 + 16x − 9b. 2x^2 + 80x + 400c. x^2 − 6x − 9d. 4x^2 − 12x + 9e. −x^2 + 14x + 49

Answer 1

x = 3/2 or 1.5

Explanation:

• All 5 of the quadratics are in standard form, whose general form is

`ax^2+bx+c`

• One of the ways in which we solve quadratic equations is through the quadratic formula which is

`x=(-b+/-√(b^2-4ac) )/(2a)`, where x is the root(s)

• We can find the total number of solutions a quadratic equation has using the discriminant from the quadratic formula which is

`b^2-4ac`

• When the discriminant is greater than 0, there is 2 distinct roots
• When the discriminant is equal to 0, there is 1 distinct root
• When the discriminant is less than 0, there are 0 distinct/"real" roots

(a.) For 4x^2 + 16x - 9b, 4 is our a value, 16 is our b value and -9 is our c value:

`16^2-4(4)(-9)\\256+144\\400 > 0`

(b.) For 2x^2 + 80x + 400, 2 is our a value, 80 is our b value, and 400 is our c value:

`80^2-4(2)(400)\\6400-3200\\3200 > 0`

(c.) For x^2 - 6x - 9, 1 is our a value, -6 is our b value and -9 is our c value

• Quick fact: for x^2 or -x^2, there's a 1 or -1 in front of the variable, but it's usually not written because it's a well known mathematical effect and it's assumed we already know this)

`(-6)^2-4(1)(-9)\\36+36\\72 > 0`

(d.) For 4x^2 - 12x + 9, 4 is our a value, -12 is our b value, and 9 is our c value:

`(-12)^2-4(4)(9)\\144-144\\0=0`

• We don't have to do (e.) because we see that since the discriminant for (d.) equals 0, this is the quadratic with only one distinct solution/root
• We can now solve for this root using the quadratic formula

`x=(-(-12)+/-√((-12)^2-4(4)(9)) )/(2(4))\\ \\x=(12+/-√(0) )/(8) \\\\x=12/8=3/2\\or\\x=1.5`