We are tasked with factoring the quadratic expression x squared minus ten x plus thirteen. Factoring means expressing a polynomial as a product of simpler polynomials. Let's start by identifying the coefficients in the general form a x squared plus b x plus c.
For a monic quadratic (where the coefficient of x squared is one), we look for two numbers p and q such that their sum equals the coefficient of x, and their product equals the constant term. In our case, we need p plus q to equal negative ten, and p times q to equal thirteen.
Let's list all integer factor pairs of thirteen and check their sums. The positive pair one times thirteen gives a sum of fourteen. The negative pair negative one times negative thirteen gives a sum of negative fourteen. Neither sum equals negative ten, so the quadratic cannot be factored using integers.
To understand why factoring failed, we can analyze the discriminant. The discriminant delta equals b squared minus four a c. Substituting our values: negative ten squared minus four times one times thirteen. This evaluates to one hundred minus fifty-two, which is forty-eight. Since forty-eight is not a perfect square, the roots are irrational, confirming that the quadratic cannot be factored into linear factors with rational coefficients.
Since factoring over integers isn't possible, let's try completing the square. We start with x squared minus ten x plus thirteen. To complete the square for x squared minus ten x, we take half of negative ten, which is negative five, and square it to get twenty-five. We add and subtract twenty-five, giving us x squared minus ten x plus twenty-five minus twenty-five plus thirteen. This simplifies to the vertex form: x minus five quantity squared minus twelve.
Now we can find the exact roots using the completed square form. Starting with x minus five quantity squared minus twelve equals zero, we add twelve to both sides to get x minus five quantity squared equals twelve. Taking the square root of both sides gives x minus five equals plus or minus the square root of twelve. Simplifying the radical, the square root of twelve is two root three. Adding five to both sides yields our exact roots: x equals five plus or minus two root three.
Let's summarize our findings. First, the quadratic cannot be factored into linear factors with integer coefficients. Second, by completing the square, we found the vertex form to be x minus five quantity squared minus twelve. Third, the exact roots are x equals five plus or minus two root three. As a verification, substituting one of these roots back into the original expression will yield zero, confirming our solution.