SOLUTION: f(x)=x^2-3 f(x)=-1/2x^2+3 f(x)=x^2+4x+1 f(x)=x^2-10x+3

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Question 102946This question is from textbook College Algebra
: f(x)=x^2-3
f(x)=-1/2x^2+3
f(x)=x^2+4x+1
f(x)=x^2-10x+3 This question is from textbook College Algebra

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
1.
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B0x%2B-3+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%280%29%5E2-4%2A1%2A-3=12.

Discriminant d=12 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-0%2B-sqrt%28+12+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%280%29%2Bsqrt%28+12+%29%29%2F2%5C1+=+1.73205080756888
x%5B2%5D+=+%28-%280%29-sqrt%28+12+%29%29%2F2%5C1+=+-1.73205080756888

Quadratic expression 1x%5E2%2B0x%2B-3 can be factored:
1x%5E2%2B0x%2B-3+=+1%28x-1.73205080756888%29%2A%28x--1.73205080756888%29
Again, the answer is: 1.73205080756888, -1.73205080756888. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B0%2Ax%2B-3+%29


2.
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case -0.5x%5E2%2B0x%2B3+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%280%29%5E2-4%2A-0.5%2A3=6.

Discriminant d=6 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-0%2B-sqrt%28+6+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%280%29%2Bsqrt%28+6+%29%29%2F2%5C-0.5+=+-2.44948974278318
x%5B2%5D+=+%28-%280%29-sqrt%28+6+%29%29%2F2%5C-0.5+=+2.44948974278318

Quadratic expression -0.5x%5E2%2B0x%2B3 can be factored:
-0.5x%5E2%2B0x%2B3+=+-0.5%28x--2.44948974278318%29%2A%28x-2.44948974278318%29
Again, the answer is: -2.44948974278318, 2.44948974278318. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-0.5%2Ax%5E2%2B0%2Ax%2B3+%29


3.
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B4x%2B1+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%284%29%5E2-4%2A1%2A1=12.

Discriminant d=12 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-4%2B-sqrt%28+12+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%284%29%2Bsqrt%28+12+%29%29%2F2%5C1+=+-0.267949192431123
x%5B2%5D+=+%28-%284%29-sqrt%28+12+%29%29%2F2%5C1+=+-3.73205080756888

Quadratic expression 1x%5E2%2B4x%2B1 can be factored:
1x%5E2%2B4x%2B1+=+1%28x--0.267949192431123%29%2A%28x--3.73205080756888%29
Again, the answer is: -0.267949192431123, -3.73205080756888. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B4%2Ax%2B1+%29


4.
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-10x%2B3+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-10%29%5E2-4%2A1%2A3=88.

Discriminant d=88 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--10%2B-sqrt%28+88+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-10%29%2Bsqrt%28+88+%29%29%2F2%5C1+=+9.69041575982343
x%5B2%5D+=+%28-%28-10%29-sqrt%28+88+%29%29%2F2%5C1+=+0.30958424017657

Quadratic expression 1x%5E2%2B-10x%2B3 can be factored:
1x%5E2%2B-10x%2B3+=+1%28x-9.69041575982343%29%2A%28x-0.30958424017657%29
Again, the answer is: 9.69041575982343, 0.30958424017657. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-10%2Ax%2B3+%29