Given a quadratic function

$\displaystyle f(x)=ax^2 +bx+c$

the vertex will always be

$\displaystyle (-\frac{b}{2a},\frac{4ac-b^2}{4a})$.

Thus, since our function is 

$\displaystyle f(x) = 2x^2 - 3x + 17$

$\displaystyle a=2$, $\displaystyle b=-3$, and $\displaystyle c=17$.

We plug these variables into the formula to get the vertex as

$\displaystyle (-\frac{-3}{2\times2},\frac{4\times2\times17-(-3)^2}{4\times2})$

$\displaystyle (-\frac{-3}{4},\frac{136-9}{8})$

$\displaystyle =(\frac{3}{4},\frac{127}{8})$.

Hence, the vertex of

$\displaystyle f(x) = 2x^2 - 3x + 17$

is

$\displaystyle (\frac{3}{4},\frac{127}{8})$.

The discriminant of a quadratic polynomial

$\displaystyle f(x)=ax^2+bx+c$

is given by

$\displaystyle b^2-4ac$.

Thus, since our quadratic polynomial is 

$\displaystyle f(x)=6x^2-3x+7$,

$\displaystyle a=6$, $\displaystyle b=-3$, and $\displaystyle c=7$. 

Plugging these values into the discriminant equation, we find that the discriminant is

$\displaystyle 9-4\times6\times7$.

A polynomial equation of the form

$\displaystyle ax^{2}+bx+c = 0$

has one and only one (real) solution if and only if its discriminant is equal to zero - that is, if its coefficients satisfy the equation

$\displaystyle b^{2}-4ac = 0$

In each of the choices, $\displaystyle a = 4$ and $\displaystyle c = 25$, so it suffices to determine the value of $\displaystyle b$ which satisfies this equation. Substituting, we get

$\displaystyle b^{2}-4 (4)(25) = 0$

$\displaystyle b^{2}-400 = 0$

Solve for $\displaystyle b$ by first adding 400 to both sides:

$\displaystyle b^{2}-400 + 400 = 0 + 400$

$\displaystyle b^{2} = 400$

Take the square root of both sides:

$\displaystyle b =\pm \sqrt{400} = \pm 20$

The choice that matches this value of $\displaystyle b$ is the equation

$\displaystyle 4x^{2}+ 20x + 25 = 0$

To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form 

$\displaystyle ax^{2}+bx+c = 0$

To accomplish this, first, multiply the binomials on the left using the FOIL technique:

$\displaystyle (2x-5)(x+4) = -6$

$\displaystyle 2x(x)+2x(4)-5(x)-5(4)= -6$

$\displaystyle 2x ^{2}+8x -5x-20= -6$

Collect like terms:

$\displaystyle 2x ^{2}+3x-20= -6$

Add 6 to both sides:

$\displaystyle 2x ^{2}+3x-20 + 6 = -6 + 6$

$\displaystyle 2x ^{2}+3x-14= 0$

The key to determining the nature of the solution set is to examine the discriminant $\displaystyle b^{2} - 4ac$. Setting $\displaystyle a = 2, b = 3, c= -14$, the value of the discriminant is 

$\displaystyle b^{2} - 4ac= 3^{2} -4(2)(-14) = 9 - (-112) = 9+112 = 121$

The discriminant is a positive number; furthermore, it is a perfect square, being equal to the square of 11. Therefore, the solution set comprises two rational solutions.

A polynomial equation of the standard form

$\displaystyle ax^{2}+bx+c = 0$

has one and only one (real) solution if and only if its discriminant is equal to zero - that is, if its coefficients satisfy the equation

$\displaystyle b^{2}-4ac = 0$

Each of the choices can be rewritten in standard form by subtracting the term on the right from both sides. One of the choices can be rewritten as follows:

$\displaystyle 9x^{2} + 25 =10x$

$\displaystyle 9x^{2} + 25 - 10x =10x - 10x$

$\displaystyle 9x^{2} - 10x + 25 =0$

By similar reasoning, the other four choices can be written:

$\displaystyle 9x^{2} - 20x + 25 =0$

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

$\displaystyle 9x^{2} - 40x + 25 =0$

$\displaystyle 9x^{2} - 50x + 25 =0$

In each of the five standard forms, $\displaystyle a= 9$ and $\displaystyle c = 25$, so it is necessary to determine the value of $\displaystyle b$ that produces a zero discriminant. Substituting accordingly:

$\displaystyle b^{2}-4ac = 0$

$\displaystyle b^{2}-4(9)(25) = 0$

$\displaystyle b^{2}-900 = 0$

Add 900 to both sides and take the square root:

$\displaystyle b^{2}-900 + 900 = 0+ 900$

$\displaystyle b^{2}= 900$

$\displaystyle b = \pm \sqrt{900} = \pm 30$

Of the five standard forms, 

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

fits this condition. This is the standard form of the equation 

$\displaystyle 9x^{2} + 25 =30x$,

the correct choice.

To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form 

$\displaystyle ax^{2}+bx+c = 0$

This can be done by simply switching the first and second terms:

$\displaystyle 6x+ 2x^{2}+ 7= 0$

$\displaystyle 2x^{2}+6x+ 7= 0$

The key to determining the nature of the solution set is to examine the discriminant  $\displaystyle b^{2} - 4ac$. Setting $\displaystyle a = 2, b= 6, c= 7$, the value of the discriminant is 

$\displaystyle b^{2} - 4ac =6^{2} - 4(2)(7) = 36-56=-20$

The discriminant has a negative value. It follows that the solution set comprises two imaginary values.

To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form 

$\displaystyle ax^{2}+bx+c = 0$

This can be done by adding 17 to both sides:

$\displaystyle 2x^{2}+ 5x+ 17 = -17+ 17$

$\displaystyle 2x^{2}+ 5x+ 17 = 0$

The key to determining the nature of the solution set is to examine the discriminant 

$\displaystyle b^{2} - 4ac$. Setting $\displaystyle a = 2, b = 5, c= 17$, the value of the discriminant is 

$\displaystyle 5^{2} - 4(2)(17) = 25 - 136 = -111$

This value is negative. Consequently, the solution set comprises two imaginary numbers.

To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form 

$\displaystyle ax^{2}+bx+c = 0$

To accomplish this, first, multiply the binomials on the left using the FOIL technique:

$\displaystyle (x-4)(x-5)= 18$

$\displaystyle x(x)-x(5)-4(x)+4(5)= 18$

$\displaystyle x^{2}-5x -4x+20= 18$

Collect like terms:

$\displaystyle x^{2}-9x+20= 18$

Now, subtract 18 from both sides:

$\displaystyle x^{2}-9x+20 -18 = 18 -18$

$\displaystyle x^{2}-9x+2 = 0$

The key to determining the nature of the solution set is to examine the discriminant $\displaystyle b^{2} - 4ac$. Setting $\displaystyle a = 1, b = -9, c= 2$, the value of the discriminant is 

$\displaystyle b^{2} - 4ac = (-9)^{2} - 4(1)(2)= 81 - 8 = 73$

The discriminant is a positive number, so there are two real solutions. Since 73 is not a perfect square, the solutions are irrational.

To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form 

$\displaystyle ax^{2}+bx+c = 0$

To accomplish this, first, multiply the binomials on the left using the FOIL technique:

$\displaystyle (x-4)(x-5)= -18$

$\displaystyle x(x)-x(5)-4(x)+4(5)=- 18$

$\displaystyle x^{2}-5x -4x+20= -18$

Collect like terms:

$\displaystyle x^{2}-9x+20=- 18$

Now, add 18 to both sides:

$\displaystyle x^{2}-9x+20 +18 = -18+18$

$\displaystyle x^{2}-9x+38 = 0$

The key to determining the nature of the solution set is to examine the discriminant $\displaystyle b^{2} - 4ac$. Setting $\displaystyle a = 1, b = -9, c= 38$, the value of the discriminant is 

$\displaystyle b^{2} - 4ac = (-9)^{2} - 4(1)(38)= 81 - 152 = -71$

This discriminant is negative. Consequently, the solution set comprises two imaginary numbers.

To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form 

$\displaystyle ax^{2}+bx+c = 0$

This can be done by switching the first and third terms on the left:

$\displaystyle 20 + 4x - x^{2} = 0$

$\displaystyle - x^{2} + 4x + 20 = 0$

The key to determining the nature of the solution set is to examine the discriminant 

$\displaystyle b^{2} - 4ac$. Setting $\displaystyle a = -1, b = 4, c=20$, the value of the discriminant is

$\displaystyle 4^{2} - 4 (-1)(20)= 16 - (-80 )= 96$.

The discriminant is a positive number but not a perfect square. Therefore, there are two irrational solutions.