Solve 750000=(x)^(2)+250x | Equatix - Math Solver

Fractions & Decimals

Ratios & Proportions

Basic Number Theory

Integers & Absolute Values

Factors & Multiples

Prime Factorization

Basic Equations & Inequalities

Coordinate Plane & Graphing

Linear Equations & Inequalities

Quadratic Equations

Polynomials & Factoring

Functions & Graphs

Systems of Equations

Trigonometric Functions

Trigonometric Identities & Equations

Right Triangle Trigonometry

Graphing Trigonometric Functions

Limits & Continuity

Derivatives & Applications

Integrals & Applications

Series & Sequences

Multivariable Calculus

Functions & Graphs

Polynomial & Rational Functions

Exponential & Logarithmic Functions

Sequences & Series

Descriptive Statistics

Normal Distribution

Hypothesis Testing

Regression Analysis

Linear Programming

Sets & Counting

Eigenvalues & Eigenvectors

Linear Transformations

Atomic Structure

Thermochemistry

Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$750000= { x }^{ 2 } +250x$$

Solve for x

$x=-1000$
$x=750$

Steps Using Factoring

Steps Using Factoring By Grouping

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using Factoring

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+250x=750000$$

Subtract $750000$ from both sides.

$$x^{2}+250x-750000=0$$

To solve the equation, factor $x^{2}+250x-750000$ using formula $x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right)$. To find $a$ and $b$, set up a system to be solved.

$$a+b=250$$ $$ab=-750000$$

Since $ab$ is negative, $a$ and $b$ have the opposite signs. Since $a+b$ is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product $-750000$.

$$-1,750000$$ $$-2,375000$$ $$-3,250000$$ $$-4,187500$$ $$-5,150000$$ $$-6,125000$$ $$-8,93750$$ $$-10,75000$$ $$-12,62500$$ $$-15,50000$$ $$-16,46875$$ $$-20,37500$$ $$-24,31250$$ $$-25,30000$$ $$-30,25000$$ $$-40,18750$$ $$-48,15625$$ $$-50,15000$$ $$-60,12500$$ $$-75,10000$$ $$-80,9375$$ $$-100,7500$$ $$-120,6250$$ $$-125,6000$$ $$-150,5000$$ $$-200,3750$$ $$-240,3125$$ $$-250,3000$$ $$-300,2500$$ $$-375,2000$$ $$-400,1875$$ $$-500,1500$$ $$-600,1250$$ $$-625,1200$$ $$-750,1000$$

Calculate the sum for each pair.

$$-1+750000=749999$$ $$-2+375000=374998$$ $$-3+250000=249997$$ $$-4+187500=187496$$ $$-5+150000=149995$$ $$-6+125000=124994$$ $$-8+93750=93742$$ $$-10+75000=74990$$ $$-12+62500=62488$$ $$-15+50000=49985$$ $$-16+46875=46859$$ $$-20+37500=37480$$ $$-24+31250=31226$$ $$-25+30000=29975$$ $$-30+25000=24970$$ $$-40+18750=18710$$ $$-48+15625=15577$$ $$-50+15000=14950$$ $$-60+12500=12440$$ $$-75+10000=9925$$ $$-80+9375=9295$$ $$-100+7500=7400$$ $$-120+6250=6130$$ $$-125+6000=5875$$ $$-150+5000=4850$$ $$-200+3750=3550$$ $$-240+3125=2885$$ $$-250+3000=2750$$ $$-300+2500=2200$$ $$-375+2000=1625$$ $$-400+1875=1475$$ $$-500+1500=1000$$ $$-600+1250=650$$ $$-625+1200=575$$ $$-750+1000=250$$

The solution is the pair that gives sum $250$.

$$a=-750$$ $$b=1000$$

Rewrite factored expression $\left(x+a\right)\left(x+b\right)$ using the obtained values.

$$\left(x-750\right)\left(x+1000\right)$$

To find equation solutions, solve $x-750=0$ and $x+1000=0$.

$$x=750$$ $$x=-1000$$

Steps Using Factoring By Grouping

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+250x=750000$$

Subtract $750000$ from both sides.

$$x^{2}+250x-750000=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as $x^{2}+ax+bx-750000$. To find $a$ and $b$, set up a system to be solved.

$$a+b=250$$ $$ab=1\left(-750000\right)=-750000$$

Since $ab$ is negative, $a$ and $b$ have the opposite signs. Since $a+b$ is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product $-750000$.

$$-1,750000$$ $$-2,375000$$ $$-3,250000$$ $$-4,187500$$ $$-5,150000$$ $$-6,125000$$ $$-8,93750$$ $$-10,75000$$ $$-12,62500$$ $$-15,50000$$ $$-16,46875$$ $$-20,37500$$ $$-24,31250$$ $$-25,30000$$ $$-30,25000$$ $$-40,18750$$ $$-48,15625$$ $$-50,15000$$ $$-60,12500$$ $$-75,10000$$ $$-80,9375$$ $$-100,7500$$ $$-120,6250$$ $$-125,6000$$ $$-150,5000$$ $$-200,3750$$ $$-240,3125$$ $$-250,3000$$ $$-300,2500$$ $$-375,2000$$ $$-400,1875$$ $$-500,1500$$ $$-600,1250$$ $$-625,1200$$ $$-750,1000$$

Calculate the sum for each pair.

$$-1+750000=749999$$ $$-2+375000=374998$$ $$-3+250000=249997$$ $$-4+187500=187496$$ $$-5+150000=149995$$ $$-6+125000=124994$$ $$-8+93750=93742$$ $$-10+75000=74990$$ $$-12+62500=62488$$ $$-15+50000=49985$$ $$-16+46875=46859$$ $$-20+37500=37480$$ $$-24+31250=31226$$ $$-25+30000=29975$$ $$-30+25000=24970$$ $$-40+18750=18710$$ $$-48+15625=15577$$ $$-50+15000=14950$$ $$-60+12500=12440$$ $$-75+10000=9925$$ $$-80+9375=9295$$ $$-100+7500=7400$$ $$-120+6250=6130$$ $$-125+6000=5875$$ $$-150+5000=4850$$ $$-200+3750=3550$$ $$-240+3125=2885$$ $$-250+3000=2750$$ $$-300+2500=2200$$ $$-375+2000=1625$$ $$-400+1875=1475$$ $$-500+1500=1000$$ $$-600+1250=650$$ $$-625+1200=575$$ $$-750+1000=250$$

The solution is the pair that gives sum $250$.

$$a=-750$$ $$b=1000$$

Rewrite $x^{2}+250x-750000$ as $\left(x^{2}-750x\right)+\left(1000x-750000\right)$.

$$\left(x^{2}-750x\right)+\left(1000x-750000\right)$$

Factor out $x$ in the first and $1000$ in the second group.

$$x\left(x-750\right)+1000\left(x-750\right)$$

Factor out common term $x-750$ by using distributive property.

$$\left(x-750\right)\left(x+1000\right)$$

To find equation solutions, solve $x-750=0$ and $x+1000=0$.

$$x=750$$ $$x=-1000$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+250x=750000$$

Subtract $750000$ from both sides.

$$x^{2}+250x-750000=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $1$ for $a$, $250$ for $b$, and $-750000$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{-250±\sqrt{250^{2}-4\left(-750000\right)}}{2}$$

Square $250$.

$$x=\frac{-250±\sqrt{62500-4\left(-750000\right)}}{2}$$

Multiply $-4$ times $-750000$.

$$x=\frac{-250±\sqrt{62500+3000000}}{2}$$

Add $62500$ to $3000000$.

$$x=\frac{-250±\sqrt{3062500}}{2}$$

Take the square root of $3062500$.

$$x=\frac{-250±1750}{2}$$

Now solve the equation $x=\frac{-250±1750}{2}$ when $±$ is plus. Add $-250$ to $1750$.

$$x=\frac{1500}{2}$$

Divide $1500$ by $2$.

$$x=750$$

Now solve the equation $x=\frac{-250±1750}{2}$ when $±$ is minus. Subtract $1750$ from $-250$.

$$x=-\frac{2000}{2}$$

Divide $-2000$ by $2$.

$$x=-1000$$

The equation is now solved.

$$x=750$$ $$x=-1000$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+250x=750000$$

Divide $250$, the coefficient of the $x$ term, by $2$ to get $125$. Then add the square of $125$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+250x+125^{2}=750000+125^{2}$$

Square $125$.

$$x^{2}+250x+15625=750000+15625$$

Add $750000$ to $15625$.

$$x^{2}+250x+15625=765625$$

Factor $x^{2}+250x+15625$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(x+125\right)^{2}=765625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+125\right)^{2}}=\sqrt{765625}$$

Simplify.

$$x+125=875$$ $$x+125=-875$$

Subtract $125$ from both sides of the equation.

$$x=750$$ $$x=-1000$$