Question 218·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

Type $y=5x^2-11x-12$. Desmos draws a curve crossing the $x$-axis on both sides of $0$. An x-intercept is a crossing where $y=0$, so each crossing's $x$-value solves the equation. The right-hand crossing has a positive $x$-value.

Add $5x_1^2-11x_1-12\sim0\{x_1>0\}$. In this regression, $x_1$ is the number Desmos finds, $\sim$ tells it to match the two sides, and the braces allow only positive values. Under PARAMETERS, Desmos shows $x_1=3$. So the positive solution is $3$. Grid in 3.

To factor is to rewrite an expression as a product. Multiply the first and last coefficients: $5(-12)=-60$. The numbers $-15$ and $4$ multiply to $-60$ and add to $-11$, so split $-11x$ using them:

Take $5x$ out of the first two terms and $4$ out of the last two. Both groups then have the same two-term expression:

The common factor $(x-3)$ appears in both groups. Take that shared factor out:

Use the zero-product property: if two numbers multiply to $0$, at least one is $0$. So set each whole factor equal to $0$:

For positive $x$, both $5x$ and $4$ are positive, so $5x+4$ cannot equal $0$. The positive solution must instead satisfy

Add $3$ to both sides: $x=3$. This is positive, so it is the solution the question asks for. Grid in 3.