Solve Quadratic equations x^2+24x-756=0 Tiger Algebra Solver (2025)

Step by step solution :

Step 1 :

Trying to factor by splitting the middle term

1.1Factoring x2+24x-756

The first term is, x2 its coefficient is 1.
The middle term is, +24x its coefficient is 24.
The last term, "the constant", is -756

Step-1 : Multiply the coefficient of the first term by the constant 1-756=-756

Step-2 : Find two factors of -756 whose sum equals the coefficient of the middle term, which is 24.

-756+1=-755
-378+2=-376
-252+3=-249
-189+4=-185
-126+6=-120
-108+7=-101
-84+9=-75
-63+12=-51
-54+14=-40
-42+18=-24
-36+21=-15
-28+27=-1
-27+28=1
-21+36=15
-18+42=24That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step2above, -18 and 42
x2 - 18x+42x - 756

Step-4 : Add up the first 2 terms, pulling out like factors:
x•(x-18)
Add up the last 2 terms, pulling out common factors:
42•(x-18)
Step-5:Add up the four terms of step4:
(x+42)•(x-18)
Which is the desired factorization

Equation at the end of step 1 :

 (x + 42) • (x - 18) = 0 

Step 2 :

Theory - Roots of a product :

2.1 A product of several terms equals zero.When a product of two or more terms equals zero, then at least one of the terms must be zero.We shall now solve each term = 0 separatelyIn other words, we are going to solve as many equations as there are terms in the productAny solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation:

2.2Solve:x+42 = 0Subtract 42 from both sides of the equation:
x = -42

Solving a Single Variable Equation:

2.3Solve:x-18 = 0Add 18 to both sides of the equation:
x = 18

Supplement : Solving Quadratic Equation Directly

Solving x2+24x-756 = 0 directly 

Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula

Parabola, Finding the Vertex:

3.1Find the Vertex ofy = x2+24x-756Parabolas have a highest or a lowest point called the Vertex.Our parabola opens up and accordingly has a lowest point (AKA absolute minimum).We know this even before plotting "y" because the coefficient of the first term,1, is positive (greater than zero).Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two x-intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex.For any parabola,Ax2+Bx+C,the x-coordinate of the vertex is given by -B/(2A). In our case the x coordinate is -12.0000Plugging into the parabola formula -12.0000 for x we can calculate the y-coordinate:
y = 1.0 * -12.00 * -12.00 + 24.0 * -12.00 - 756.0
or y = -900.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for : y = x2+24x-756
Axis of Symmetry (dashed) {x}={-12.00}
Vertex at {x,y} = {-12.00,-900.00}
x-Intercepts (Roots) :
Root 1 at {x,y} = {-42.00, 0.00}
Root 2 at {x,y} = {18.00, 0.00}

Solve Quadratic Equation by Completing The Square

3.2Solvingx2+24x-756 = 0 by Completing The Square.Add 756 to both side of the equation :
x2+24x = 756

Now the clever bit: Take the coefficient of x, which is 24, divide by two, giving 12, and finally square it giving 144

Add 144 to both sides of the equation :
On the right hand side we have:
756+144or, (756/1)+(144/1)
The common denominator of the two fractions is 1Adding (756/1)+(144/1) gives 900/1
So adding to both sides we finally get:
x2+24x+144 = 900

Adding 144 has completed the left hand side into a perfect square :
x2+24x+144=
(x+12)(x+12)=
(x+12)2
Things which are equal to the same thing are also equal to one another. Since
x2+24x+144 = 900 and
x2+24x+144 = (x+12)2
then, according to the law of transitivity,
(x+12)2 = 900

We'll refer to this Equation as Eq. #3.2.1

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of
(x+12)2 is
(x+12)2/2=
(x+12)1=
x+12

Now, applying the Square Root Principle to Eq.#3.2.1 we get:
x+12= 900

Subtract 12 from both sides to obtain:
x = -12 + √ 900

Since a square root has two values, one positive and the other negative
x2 + 24x - 756 = 0
has two solutions:
x = -12 + √ 900
or
x = -12 - √ 900

Solve Quadratic Equation using the Quadratic Formula

3.3Solvingx2+24x-756 = 0 by the Quadratic Formula.According to the Quadratic Formula,x, the solution forAx2+Bx+C= 0 , where A, B and C are numbers, often called coefficients, is given by :

-B± √B2-4AC
x = ————————
2A
In our case,A= 1
B= 24
C=-756
Accordingly,B2-4AC=
576 - (-3024) =
3600
Applying the quadratic formula :

-24 ± √ 3600
x=———————
2
Can 3600 be simplified ?

Yes!The prime factorization of 3600is
2•2•2•2•3•3•5•5
To be able to remove something from under the radical, there have to be 2 instances of it (because we are taking a square i.e. second root).

3600 =√2•2•2•2•3•3•5•5 =2•2•3•5•√ 1 =
±60 •√ 1 =
±60

So now we are looking at:
x=(-24±60)/2

Two real solutions:

x =(-24+√3600)/2=-12+30= 18.000

or:

x =(-24-√3600)/2=-12-30= -42.000

Two solutions were found :

  1. x = 18
  2. x = -42
Solve Quadratic equations x^2+24x-756=0 Tiger Algebra Solver (2025)

FAQs

How is the quadratic formula derived? ›

The quadratic formula, x=- b±sqrt(b^2-4ac)/2a, can be derived by completing the square on the general standard form a quadratic equation. Recall that completing the square is a method for solving quadratic equations.

How to solve quadratic equations without the quadratic formula? ›

Use the square root method! If the only variable in your equation is x², move all of the other numbers to the other side of the equation. Then, find the square root of both sides of the equation.

What are the different methods of solving quadratic equations? ›

Answer: There are various methods by which you can solve a quadratic equation such as: factorization, completing the square, quadratic formula, and graphing. These are the four general methods by which we can solve a quadratic equation.

How to do a quadratic formula step by step? ›

Applying the Quadratic Formula

Step 1: Identify a, b, and c in the quadratic equation a x 2 + b x + c = 0 . Step 2: Substitute the values from step 1 into the quadratic formula x = − b ± b 2 − 4 a c 2 a . Step 3: Simplify, making sure to follow the order of operations.

How to find the roots of a quadratic equation? ›

Important Formulas for Quadratic Equation Roots include:

ax² + bx + c = 0 is a quadratic equation. Use the formula x = (-b ± √ (b² – 4ac) )/2a. to calculate the roots. D = b² – 4ac is the discriminant.

Does the quadratic formula give you the roots? ›

For a quadratic equation ax2 + bx + c = 0, The roots are calculated using the formula, x = (-b ± √ (b2 - 4ac) )/2a. Discriminant is, D = b2 - 4ac. If D > 0, then the equation has two real and distinct roots.

What quadratic equation Cannot be solved by factoring? ›

Not every quadratic equation can be solved by factoring or by extraction of roots. For example, the expression x 2 + x − 1 cannot be factored, so the equation x 2 + x − 1 = 0 cannot be solved by factoring. For other equations, factoring may be difficult.

What is the Almighty formula for the quadratic equation? ›

The quadratic equation in its standard form is ax2 + bx + c = 0, where a and b are the coefficients, x is the variable, and c is the constant term. The important condition for an equation to be a quadratic equation is the coefficient of x2 is a non-zero term (a ≠ 0).

How can you determine the zeros, the vertex, and axis of symmetry of a quadratic graph? ›

1 Expert Answer

We can find the zeros, axis of symmetry, and vertex by putting the functions in vertex form. in vertex form, y = 3(x2 + 5x + (25/4)) - (3/4) the term in bold parenthesis is the squared term. The axis of symmetry is the x coordinate of the vertex.

When should you complete the square? ›

If you are trying to find the roots of a quadratic equation, then completing the square will 'always work', in the sense that it does not require the factors to be rational and in the sense that it will give you the complex roots if the quadratic's roots are not real.

How to find the equation of a quadratic function? ›

A quadratic function, of the form f(x) = ax2 + bx + c, is determined by three points. Given three points on the graph of a quadratic function, we can work out the function by finding a, b and c algebraically. This will require solving a system of three equations in three unknowns.

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